1 00:00:00,030 --> 00:00:02,480 this video is sponsored by dash lane 2 00:00:02,480 --> 00:00:04,950 circuits and electronics image 3 00:00:04,950 --> 00:00:07,649 processing computer graphics quantum 4 00:00:07,649 --> 00:00:09,900 mechanics the Google page rank algorithm 5 00:00:09,900 --> 00:00:13,710 any kind of network other stuff these 6 00:00:13,710 --> 00:00:15,269 are the kinds of things where matrices 7 00:00:15,269 --> 00:00:17,399 are used and extremely important for 8 00:00:17,399 --> 00:00:18,719 understanding or analyzing different 9 00:00:18,719 --> 00:00:21,000 systems and although I can't discuss 10 00:00:21,000 --> 00:00:23,189 everything in one video this should give 11 00:00:23,189 --> 00:00:24,778 you some insight into the applications 12 00:00:24,778 --> 00:00:26,609 of matrices beyond an introductory 13 00:00:26,609 --> 00:00:29,609 course might include now in the 14 00:00:29,609 --> 00:00:31,170 beginning matrices can be one of the 15 00:00:31,170 --> 00:00:32,850 most boring subjects we learn in math 16 00:00:32,850 --> 00:00:34,920 maybe now for everyone but least that's 17 00:00:34,920 --> 00:00:37,320 how it was for me I mean we're told hey 18 00:00:37,320 --> 00:00:39,210 here's how matrix addition works real 19 00:00:39,210 --> 00:00:40,799 simply just some of the corresponding 20 00:00:40,799 --> 00:00:43,609 entries and you have your answer 21 00:00:43,609 --> 00:00:46,049 then multiply a matrix by a single 22 00:00:46,049 --> 00:00:48,179 number is as simple as multiplying every 23 00:00:48,179 --> 00:00:52,378 entry by that value but when it comes to 24 00:00:52,378 --> 00:00:54,600 matrix multiplication we do this weird 25 00:00:54,600 --> 00:00:57,299 row by column dot product multiplication 26 00:00:57,299 --> 00:00:59,909 which some teachers just give no context 27 00:00:59,909 --> 00:01:02,698 to so yeah this isn't the kind of stuff 28 00:01:02,698 --> 00:01:04,438 that makes you in a major in matrix math 29 00:01:04,438 --> 00:01:07,109 anytime soon I mean you might learn more 30 00:01:07,109 --> 00:01:08,728 in high school but overall a lot of it 31 00:01:08,728 --> 00:01:10,950 just isn't that exciting however I 32 00:01:10,950 --> 00:01:13,379 promise matrices are used way more than 33 00:01:13,379 --> 00:01:15,420 you probably think but the first thing 34 00:01:15,420 --> 00:01:17,879 we need to realize is that matrices do 35 00:01:17,879 --> 00:01:20,310 things two vectors don't take this as a 36 00:01:20,310 --> 00:01:22,650 definition cuz it's obviously not but we 37 00:01:22,650 --> 00:01:23,879 do need to see what happens when we 38 00:01:23,879 --> 00:01:27,450 multiply a matrix by a vector for 39 00:01:27,450 --> 00:01:29,009 example a vector that starts at the 40 00:01:29,009 --> 00:01:31,739 origin and ends at 1 comma 1 can be 41 00:01:31,739 --> 00:01:33,599 written in matrix form as shown X 42 00:01:33,599 --> 00:01:35,728 component on the top and Y component on 43 00:01:35,728 --> 00:01:38,188 the bottom and when you multiply by a 44 00:01:38,188 --> 00:01:40,560 2x2 matrix like this through the 45 00:01:40,560 --> 00:01:42,629 multiplication rules we get a new vector 46 00:01:42,629 --> 00:01:46,920 out in this case of 1 comma 3 so we put 47 00:01:46,920 --> 00:01:49,590 a vector in and the matrix scaled and 48 00:01:49,590 --> 00:01:53,489 rotate it to get a new vector out this 49 00:01:53,489 --> 00:01:55,228 is what I mean by the matrix doing 50 00:01:55,228 --> 00:01:57,180 things to the vector and in this case 51 00:01:57,180 --> 00:01:58,950 different inputs will be rotated and 52 00:01:58,950 --> 00:02:00,390 scaled differently which we'll see in a 53 00:02:00,390 --> 00:02:03,420 sec now some matrices are much simpler 54 00:02:03,420 --> 00:02:06,090 like this one here just rotates put a 55 00:02:06,090 --> 00:02:08,219 vector in aka multiplied by the matrix 56 00:02:08,219 --> 00:02:10,169 and out will come the same vector 57 00:02:10,169 --> 00:02:13,169 rotated 90 degrees counterclockwise 58 00:02:13,169 --> 00:02:15,629 this matrix on the other hand will just 59 00:02:15,629 --> 00:02:18,240 scale any vector that goes in comes out 60 00:02:18,240 --> 00:02:21,719 twice as long but most two-by-two 61 00:02:21,719 --> 00:02:23,219 matrices like this when we were 62 00:02:23,219 --> 00:02:25,740 analyzing aren't as simple different 63 00:02:25,740 --> 00:02:27,479 input vectors I'll just put a few here 64 00:02:27,479 --> 00:02:29,449 as an example gets scaled and rotated 65 00:02:29,449 --> 00:02:32,580 differently however the transformations 66 00:02:32,580 --> 00:02:35,189 are all linear as in any vector on the 67 00:02:35,189 --> 00:02:37,949 same line as one of those inputs will be 68 00:02:37,949 --> 00:02:39,930 mapped to a vector on the same line as 69 00:02:39,930 --> 00:02:42,810 the corresponding outputs these linear 70 00:02:42,810 --> 00:02:44,489 transformations are why we called the 71 00:02:44,489 --> 00:02:47,039 first in-depth class on matrices linear 72 00:02:47,039 --> 00:02:50,280 algebra anyways I'm going to redo those 73 00:02:50,280 --> 00:02:52,199 transformations once more but this time 74 00:02:52,199 --> 00:02:55,919 pay attention to this vector here you'll 75 00:02:55,919 --> 00:02:57,689 notice it's the only one that is just 76 00:02:57,689 --> 00:03:00,659 scaled it doesn't rotate at all and this 77 00:03:00,659 --> 00:03:02,310 would happen to any vector on that same 78 00:03:02,310 --> 00:03:05,009 line because of what we just saw any 79 00:03:05,009 --> 00:03:07,319 vector that is only scaled by a matrix 80 00:03:07,319 --> 00:03:09,509 is called an eigen vector of that matrix 81 00:03:09,509 --> 00:03:12,539 and how much the vector is scaled by or 82 00:03:12,539 --> 00:03:13,919 two in this case since the length 83 00:03:13,919 --> 00:03:17,400 doubled is known as the eigen value I'm 84 00:03:17,400 --> 00:03:18,719 not going to go through how to solve for 85 00:03:18,719 --> 00:03:20,280 these but the vocabulary will come up 86 00:03:20,280 --> 00:03:23,159 later now the last thing to mention is 87 00:03:23,159 --> 00:03:24,719 that the first application of matrices 88 00:03:24,719 --> 00:03:26,819 we typically learn is how they help us 89 00:03:26,819 --> 00:03:28,819 solve systems of equations the 90 00:03:28,819 --> 00:03:31,829 coefficients can go into a matrix the 91 00:03:31,829 --> 00:03:33,689 variables go into another and the 92 00:03:33,689 --> 00:03:36,599 outputs go here notice I'm using the 93 00:03:36,599 --> 00:03:38,219 same matrix as the one from the last 94 00:03:38,219 --> 00:03:41,310 slide by the way using the rules of 95 00:03:41,310 --> 00:03:42,989 matrix multiplication you can see that 96 00:03:42,989 --> 00:03:46,590 this and this are the exact same so 97 00:03:46,590 --> 00:03:48,840 really what this is asking is which 98 00:03:48,840 --> 00:03:52,769 input vector does this matrix map to 1 99 00:03:52,769 --> 00:03:56,129 comma 3 well we already saw the answer 100 00:03:56,129 --> 00:03:59,879 to this 1 comma 3 is this output and the 101 00:03:59,879 --> 00:04:01,829 question of which vector will the matrix 102 00:04:01,829 --> 00:04:04,500 map to this involves us just doing the 103 00:04:04,500 --> 00:04:06,389 same transformations as before but 104 00:04:06,389 --> 00:04:08,849 backwards to find the answer is 1 comma 105 00:04:08,849 --> 00:04:12,810 1 that will be our solution going 106 00:04:12,810 --> 00:04:14,459 backwards is like applying an inverse 107 00:04:14,459 --> 00:04:16,738 matrix and when the desired output is 108 00:04:16,738 --> 00:04:19,048 the one being multiplied then the vector 109 00:04:19,048 --> 00:04:22,379 it came from comes out so x equals 1 and 110 00:04:22,379 --> 00:04:25,500 y equals 1 are the solutions if we plug 111 00:04:25,500 --> 00:04:26,300 those into the 112 00:04:26,300 --> 00:04:28,129 some equations than both are satisfied 113 00:04:28,129 --> 00:04:30,168 which is exactly what we were looking 114 00:04:30,168 --> 00:04:34,430 for if you haven't seen three blue one 115 00:04:34,430 --> 00:04:36,168 Browns essence of linear algebra series 116 00:04:36,168 --> 00:04:38,120 definitely check that out this will make 117 00:04:38,120 --> 00:04:40,490 a lot more sense but here we're focused 118 00:04:40,490 --> 00:04:42,019 on applications which we're going to get 119 00:04:42,019 --> 00:04:45,228 to now now in systems of equations get 120 00:04:45,228 --> 00:04:47,449 more complex all we have to do is expand 121 00:04:47,449 --> 00:04:49,339 our matrix and we can analyze the system 122 00:04:49,339 --> 00:04:52,098 with as many variables as we want the 123 00:04:52,098 --> 00:04:54,259 reason matrices are used in circuits and 124 00:04:54,259 --> 00:04:56,389 electronics for example is because these 125 00:04:56,389 --> 00:04:58,490 can be represented by linear equations 126 00:04:58,490 --> 00:05:00,228 in which all the voltages and currents 127 00:05:00,228 --> 00:05:03,319 are the unknown variables when the 128 00:05:03,319 --> 00:05:05,120 circuits get hectic where we don't want 129 00:05:05,120 --> 00:05:06,918 to solve it by hand we can just have a 130 00:05:06,918 --> 00:05:08,930 computer find an inverse matrix and 131 00:05:08,930 --> 00:05:11,718 we'll have our currents and voltages but 132 00:05:11,718 --> 00:05:13,668 that's still not too exciting so what 133 00:05:13,668 --> 00:05:15,709 about a system that continuously evolves 134 00:05:15,709 --> 00:05:18,199 over time like for example let's say 135 00:05:18,199 --> 00:05:19,848 there's a zombie outbreak at the local 136 00:05:19,848 --> 00:05:20,538 high school 137 00:05:20,538 --> 00:05:22,430 pretty standard situation and the place 138 00:05:22,430 --> 00:05:24,649 is quarantined so no one can go in or 139 00:05:24,649 --> 00:05:26,449 out but the zombie infection is 140 00:05:26,449 --> 00:05:29,240 spreading so we've got humans in the 141 00:05:29,240 --> 00:05:31,430 school and zombies but no one is coming 142 00:05:31,430 --> 00:05:33,348 in or out so the population remains the 143 00:05:33,348 --> 00:05:37,038 same now let's say every hour 20% of 144 00:05:37,038 --> 00:05:38,810 humans will turn into zombies due to 145 00:05:38,810 --> 00:05:41,240 being infected there is a cure for the 146 00:05:41,240 --> 00:05:43,158 disease luckily however it's not always 147 00:05:43,158 --> 00:05:44,870 guaranteed to work so we'll say that 148 00:05:44,870 --> 00:05:47,360 every hour 10% of the zombies will 149 00:05:47,360 --> 00:05:50,718 return back to humans at this moment if 150 00:05:50,718 --> 00:05:54,168 there are 150 zombies and 150 humans the 151 00:05:54,168 --> 00:05:56,478 question is what is going to happen in 152 00:05:56,478 --> 00:05:59,930 the long run now we're going to assume 153 00:05:59,930 --> 00:06:01,699 the changes happen in discrete intervals 154 00:06:01,699 --> 00:06:04,218 at the hour so let's see what happens in 155 00:06:04,218 --> 00:06:07,668 the first hour for the humans of the 150 156 00:06:07,668 --> 00:06:09,860 starting out 80% of them are going to 157 00:06:09,860 --> 00:06:12,649 stay human or not become infected but we 158 00:06:12,649 --> 00:06:15,589 also have to add the 10% of the 150 159 00:06:15,589 --> 00:06:17,778 zombies that become cured and turned 160 00:06:17,778 --> 00:06:18,709 back to human 161 00:06:18,709 --> 00:06:20,569 this leaves us with a hundred and 162 00:06:20,569 --> 00:06:22,490 thirty-five humans after that first hour 163 00:06:22,490 --> 00:06:26,240 it went down just a bit for the new 164 00:06:26,240 --> 00:06:28,459 total of zombies we would take the 20% 165 00:06:28,459 --> 00:06:31,370 of humans that got infected plus the 90% 166 00:06:31,370 --> 00:06:34,129 of the 150 zombies that are not cured 167 00:06:34,129 --> 00:06:37,879 giving us a total of of course 165 since 168 00:06:37,879 --> 00:06:39,990 these two numbers add together muster 169 00:06:39,990 --> 00:06:44,790 300 but we want to know what happens 170 00:06:44,790 --> 00:06:46,379 after a long time so we got to keep 171 00:06:46,379 --> 00:06:49,110 going after another hour we write the 172 00:06:49,110 --> 00:06:50,970 same percentages except now the number 173 00:06:50,970 --> 00:06:54,329 of humans is 135 instead of 150 and for 174 00:06:54,329 --> 00:06:56,629 the zombies we got 165 instead of 150 175 00:06:56,629 --> 00:07:00,839 this now puts 125 for the humans and 175 176 00:07:00,839 --> 00:07:02,819 for the zombies so we seem to keep 177 00:07:02,819 --> 00:07:06,110 losing humans but will this continue 178 00:07:06,110 --> 00:07:08,399 well what we have here is some linear 179 00:07:08,399 --> 00:07:10,199 equations that can be represented as a 180 00:07:10,199 --> 00:07:12,600 matrix of those percentages which don't 181 00:07:12,600 --> 00:07:15,269 change this is multiplied by the inputs 182 00:07:15,269 --> 00:07:17,610 hmz or the current population of humans 183 00:07:17,610 --> 00:07:20,040 and zombies at any time and all of this 184 00:07:20,040 --> 00:07:22,170 equals the populations after that given 185 00:07:22,170 --> 00:07:25,019 our this is called a markov matrix by 186 00:07:25,019 --> 00:07:27,029 the way since its column sum to 1 and it 187 00:07:27,029 --> 00:07:29,730 has no negative values but this is like 188 00:07:29,730 --> 00:07:32,189 what we just saw the matrix we have is 189 00:07:32,189 --> 00:07:34,889 going to do stuff to or scale and rotate 190 00:07:34,889 --> 00:07:39,269 the input vector the first input was 150 191 00:07:39,269 --> 00:07:41,189 comma 150 the initial human and zombie 192 00:07:41,189 --> 00:07:43,920 populations and after 1 hour or 193 00:07:43,920 --> 00:07:46,769 multiplication it gets moved to 135 194 00:07:46,769 --> 00:07:50,100 comma 165 but we have to keep going and 195 00:07:50,100 --> 00:07:52,019 apply another transformation sending it 196 00:07:52,019 --> 00:07:55,259 to 125 comma 175 the populations we 197 00:07:55,259 --> 00:07:58,110 found after 2 hours so as we keep 198 00:07:58,110 --> 00:08:00,300 applying these multiplications the real 199 00:08:00,300 --> 00:08:03,139 question is where does this vector go 200 00:08:03,139 --> 00:08:05,459 well let me put a few vectors on the 201 00:08:05,459 --> 00:08:07,290 graph to show this each representing 202 00:08:07,290 --> 00:08:10,620 populations which add to 300 if we do 203 00:08:10,620 --> 00:08:12,269 the matrix multiplication and look at 204 00:08:12,269 --> 00:08:14,189 the transformations you'll notice this 205 00:08:14,189 --> 00:08:16,769 vector or anything on this line stays 206 00:08:16,769 --> 00:08:19,439 put while everything else moves towards 207 00:08:19,439 --> 00:08:23,069 it that vector is an eigenvector of our 208 00:08:23,069 --> 00:08:25,949 matrix the associated eigenvalue is 1 209 00:08:25,949 --> 00:08:29,189 since it doesn't scale and since that 210 00:08:29,189 --> 00:08:31,980 vector doesn't rotate or scale it is the 211 00:08:31,980 --> 00:08:33,750 equilibrium of the system and therefore 212 00:08:33,750 --> 00:08:36,659 the answer to our question after a long 213 00:08:36,659 --> 00:08:38,820 time the populations will settle to 214 00:08:38,820 --> 00:08:41,190 numbers which lie on this line and add 215 00:08:41,190 --> 00:08:44,129 to 300 which would be 100 for the humans 216 00:08:44,129 --> 00:08:46,830 and 200 for the zombies any other 217 00:08:46,830 --> 00:08:48,570 population values we'll just move a 218 00:08:48,570 --> 00:08:52,320 little closer to these after each hour 219 00:08:52,320 --> 00:08:54,028 if you put those values inside the 220 00:08:54,028 --> 00:08:56,129 equations from before you'll see the 221 00:08:56,129 --> 00:08:59,159 output remains 100 and 200 for the 222 00:08:59,159 --> 00:09:02,940 humans and zombies respectively then if 223 00:09:02,940 --> 00:09:04,828 the percentages were to change the 224 00:09:04,828 --> 00:09:06,778 question just comes down to what is the 225 00:09:06,778 --> 00:09:09,750 eigenvector of the new matrix there may 226 00:09:09,750 --> 00:09:11,879 not be a zombie infestation anytime soon 227 00:09:11,879 --> 00:09:13,620 but this kind of math could be used to 228 00:09:13,620 --> 00:09:15,419 analyze how a virus will spread 229 00:09:15,419 --> 00:09:17,339 throughout a population for example and 230 00:09:17,339 --> 00:09:19,620 one of my favorite applications of this 231 00:09:19,620 --> 00:09:21,899 is the Google page rank algorithm which 232 00:09:21,899 --> 00:09:24,059 involves Markov matrices and ranks 233 00:09:24,059 --> 00:09:26,100 websites by treating outgoing links as 234 00:09:26,100 --> 00:09:28,259 probabilities of transitioning from one 235 00:09:28,259 --> 00:09:31,169 site to another for more on that I have 236 00:09:31,169 --> 00:09:33,769 a dedicated video which I'll link below 237 00:09:33,769 --> 00:09:36,929 now moving on here's a happy story not 238 00:09:36,929 --> 00:09:40,379 at all on April 29 1992 a man named 239 00:09:40,379 --> 00:09:42,389 Reginald Denny was beaten nearly to 240 00:09:42,389 --> 00:09:45,600 death live on national TV and this was 241 00:09:45,600 --> 00:09:47,278 just a completely innocent man who had 242 00:09:47,278 --> 00:09:50,309 done nothing wrong you can see Reginald 243 00:09:50,309 --> 00:09:52,019 laying here probably unconscious after 244 00:09:52,019 --> 00:09:55,350 the attack the attack itself can be seen 245 00:09:55,350 --> 00:09:57,240 here on YouTube but in an attempt to 246 00:09:57,240 --> 00:09:59,009 knock an age-restricted like that video 247 00:09:59,009 --> 00:10:00,778 is I'm only showing the portion right 248 00:10:00,778 --> 00:10:03,509 after now the back story here is that 249 00:10:03,509 --> 00:10:06,539 April 29th 92 was the first day of the 250 00:10:06,539 --> 00:10:08,629 Rodney King riots in Los Angeles 251 00:10:08,629 --> 00:10:11,278 Reginald Denny was a truck driver whose 252 00:10:11,278 --> 00:10:13,019 route for the day involved going through 253 00:10:13,019 --> 00:10:14,789 an area where rioting was taking place 254 00:10:14,789 --> 00:10:17,610 which he was not informed about when he 255 00:10:17,610 --> 00:10:19,259 got there he was stopped by rioters 256 00:10:19,259 --> 00:10:21,509 dragged out of his truck and that's when 257 00:10:21,509 --> 00:10:24,690 the beating took place now identifying 258 00:10:24,690 --> 00:10:26,698 who assaulted Denny was not easy since 259 00:10:26,698 --> 00:10:28,318 the quality of the live footage wasn't 260 00:10:28,318 --> 00:10:30,559 amazing but what help law enforcement 261 00:10:30,559 --> 00:10:32,519 confidently identify one of the 262 00:10:32,519 --> 00:10:36,389 attackers with some advanced math to 263 00:10:36,389 --> 00:10:38,159 understand how this was accomplished we 264 00:10:38,159 --> 00:10:40,019 need to first look at what a digital 265 00:10:40,019 --> 00:10:42,899 images a digital image when you look 266 00:10:42,899 --> 00:10:45,028 real closely is just made up of a bunch 267 00:10:45,028 --> 00:10:47,850 of pixels each of a single color those 268 00:10:47,850 --> 00:10:49,470 colors can then be represented by some 269 00:10:49,470 --> 00:10:52,470 numerical value which means like a 270 00:10:52,470 --> 00:10:54,149 square picture made up of a million 271 00:10:54,149 --> 00:10:56,938 pixels 1000 on each edge could be 272 00:10:56,938 --> 00:10:59,278 represented by a thousand by 1,000 273 00:10:59,278 --> 00:11:01,318 matrix where the entries are the color 274 00:11:01,318 --> 00:11:03,230 values of each pixel 275 00:11:03,230 --> 00:11:05,090 working with black-and-white pictures is 276 00:11:05,090 --> 00:11:06,740 much easier though because the black 277 00:11:06,740 --> 00:11:08,330 pixel come you represented with a zero 278 00:11:08,330 --> 00:11:11,299 and a white can be a 1 let's say well 279 00:11:11,299 --> 00:11:13,159 actually work with grayscale images here 280 00:11:13,159 --> 00:11:15,139 though meaning anything in between zero 281 00:11:15,139 --> 00:11:17,389 and one can exist which will correspond 282 00:11:17,389 --> 00:11:20,389 to a different shade of grey so when it 283 00:11:20,389 --> 00:11:21,740 comes to image processing and 284 00:11:21,740 --> 00:11:23,750 manipulation whether it be blurring an 285 00:11:23,750 --> 00:11:26,750 image detecting edges sharpening an 286 00:11:26,750 --> 00:11:29,240 image and so on it all comes down to 287 00:11:29,240 --> 00:11:31,070 manipulating the pixels in a very 288 00:11:31,070 --> 00:11:34,009 specific way to see an example let's 289 00:11:34,009 --> 00:11:35,929 mathematically blur this image of the 290 00:11:35,929 --> 00:11:39,620 number 1 to do so I'm going to make a 3 291 00:11:39,620 --> 00:11:42,139 by 3 matrix where every entry is 1/9 292 00:11:42,139 --> 00:11:44,870 this is known as a kernel and image 293 00:11:44,870 --> 00:11:47,360 processing by the way then what we're 294 00:11:47,360 --> 00:11:49,460 going to do is lay this kernel over our 295 00:11:49,460 --> 00:11:52,070 image matrix and multiply the individual 296 00:11:52,070 --> 00:11:54,649 entries in each square together then add 297 00:11:54,649 --> 00:11:57,409 the results in this case it's just 1 298 00:11:57,409 --> 00:12:00,529 times 1/9 nine times so the sum of all 299 00:12:00,529 --> 00:12:03,200 those is one yes this kernel is really 300 00:12:03,200 --> 00:12:04,700 just finding the average of the pixels 301 00:12:04,700 --> 00:12:08,000 inside it from there we're going to take 302 00:12:08,000 --> 00:12:10,909 that sum of one and set the center pixel 303 00:12:10,909 --> 00:12:13,730 to that color in the new image it just 304 00:12:13,730 --> 00:12:15,740 so happened not to change it's still 1 305 00:12:15,740 --> 00:12:17,419 or white but that won't always be the 306 00:12:17,419 --> 00:12:20,299 case now you'll notice this grid on the 307 00:12:20,299 --> 00:12:21,950 right where the blurred image will go is 308 00:12:21,950 --> 00:12:24,769 the same size as the one on the left to 309 00:12:24,769 --> 00:12:26,720 get the entire blurred image we're just 310 00:12:26,720 --> 00:12:28,429 going to sweep that red section across 311 00:12:28,429 --> 00:12:30,889 the original one when we slide it over 312 00:12:30,889 --> 00:12:33,919 once all those pixels are still 1 so the 313 00:12:33,919 --> 00:12:35,840 average is also 1 and that's where this 314 00:12:35,840 --> 00:12:38,870 new pixel becomes but after sliding over 315 00:12:38,870 --> 00:12:41,419 again the kernel contains a black pixel 316 00:12:41,419 --> 00:12:44,059 so we find the average of 8 ones and a 317 00:12:44,059 --> 00:12:46,120 zero which is about point eight nine 318 00:12:46,120 --> 00:12:48,799 this corresponds to a very light shade 319 00:12:48,799 --> 00:12:50,480 of gray which we will put in that middle 320 00:12:50,480 --> 00:12:54,019 square after sweeping across the image 321 00:12:54,019 --> 00:12:55,879 mapping each new number to the blurred 322 00:12:55,879 --> 00:12:59,720 image these will be the values I know 323 00:12:59,720 --> 00:13:01,100 this method doesn't really account for 324 00:13:01,100 --> 00:13:03,110 the border but for our purposes we're 325 00:13:03,110 --> 00:13:05,690 just going to keep that white now I'll 326 00:13:05,690 --> 00:13:07,460 actually color and the pixels based on 327 00:13:07,460 --> 00:13:09,620 their values and we get a blurred image 328 00:13:09,620 --> 00:13:11,409 of the number one 329 00:13:11,409 --> 00:13:13,418 actually this is extremely blurred 330 00:13:13,418 --> 00:13:15,639 almost beyond recognition but if we put 331 00:13:15,639 --> 00:13:17,918 an outline along the colored region we 332 00:13:17,918 --> 00:13:20,038 can see the one is still kind of there 333 00:13:20,038 --> 00:13:22,568 the reason this is so blurred is because 334 00:13:22,568 --> 00:13:24,339 we're only working with a hundred pixels 335 00:13:24,339 --> 00:13:27,159 but what the kernel did was kind of took 336 00:13:27,159 --> 00:13:28,839 this sharp edge in the original image 337 00:13:28,839 --> 00:13:30,129 between black and white 338 00:13:30,129 --> 00:13:33,188 and smooth or averaged it so we get this 339 00:13:33,188 --> 00:13:35,259 fading from dark to light in the blurred 340 00:13:35,259 --> 00:13:39,129 image the kernel we use represents a 341 00:13:39,129 --> 00:13:41,139 type of blur known as a box blur and 342 00:13:41,139 --> 00:13:43,208 from Wikipedia if you input a picture 343 00:13:43,208 --> 00:13:45,249 with many more pixels then apply the 344 00:13:45,249 --> 00:13:48,399 blur this is the output you get but 345 00:13:48,399 --> 00:13:49,778 there are several other kernels that'll 346 00:13:49,778 --> 00:13:51,658 all accomplish different things a 347 00:13:51,658 --> 00:13:54,220 Gaussian blur also of course blurs the 348 00:13:54,220 --> 00:13:56,048 image but it assigns more weight to the 349 00:13:56,048 --> 00:13:58,808 middle square so dark pixels stay fairly 350 00:13:58,808 --> 00:14:01,389 dark and vice versa there's a sharpen 351 00:14:01,389 --> 00:14:04,119 kernel and there's edge detection 352 00:14:04,119 --> 00:14:06,220 kernels which search for sharp changes 353 00:14:06,220 --> 00:14:09,249 in color you'll notice that all the 354 00:14:09,249 --> 00:14:11,438 numbers in this kernel sum to zero so if 355 00:14:11,438 --> 00:14:13,298 we put it over a section of an image 356 00:14:13,298 --> 00:14:15,239 where all colors are roughly the same 357 00:14:15,239 --> 00:14:17,558 multiplying by these numbers then adding 358 00:14:17,558 --> 00:14:19,720 the results would just yield zero or a 359 00:14:19,720 --> 00:14:21,578 black pixel which is why that 360 00:14:21,578 --> 00:14:24,308 corresponding area is black the only 361 00:14:24,308 --> 00:14:26,470 sections that aren't are where we find 362 00:14:26,470 --> 00:14:30,928 sharp changes in color aka an edge as 363 00:14:30,928 --> 00:14:33,249 another example of edge detection here's 364 00:14:33,249 --> 00:14:35,139 a poorly taken photograph of someone's 365 00:14:35,139 --> 00:14:37,269 arm and by using edge detection 366 00:14:37,269 --> 00:14:38,918 algorithms researchers were able to 367 00:14:38,918 --> 00:14:40,778 identify a region of some kind of 368 00:14:40,778 --> 00:14:43,568 birthmark or tattoo well this is 369 00:14:43,568 --> 00:14:45,249 actually a zoomed in portion of this 370 00:14:45,249 --> 00:14:47,259 image where those men can be seen 371 00:14:47,259 --> 00:14:49,720 beating Reginald any using image 372 00:14:49,720 --> 00:14:51,249 processing techniques similar to what 373 00:14:51,249 --> 00:14:53,109 we've seen one company was able to 374 00:14:53,109 --> 00:14:54,908 determine that this mark was a rose 375 00:14:54,908 --> 00:14:57,428 tattoo affiliated with a certain gang in 376 00:14:57,428 --> 00:14:59,528 Los Angeles and it was this that helped 377 00:14:59,528 --> 00:15:01,359 him eventually secure a conviction of 378 00:15:01,359 --> 00:15:04,948 one of the perpetrators of the attack 379 00:15:05,639 --> 00:15:08,080 now all this may not have involved much 380 00:15:08,080 --> 00:15:10,419 matrix math like we saw earlier but no I 381 00:15:10,419 --> 00:15:12,070 did simplify some things to avoid going 382 00:15:12,070 --> 00:15:14,529 in too much detail and not only image 383 00:15:14,529 --> 00:15:16,720 processing but computer graphics heavily 384 00:15:16,720 --> 00:15:19,360 use matrices with these what we can do 385 00:15:19,360 --> 00:15:21,759 is take geometric data and incorporate 386 00:15:21,759 --> 00:15:24,159 it into a coordinate system we can then 387 00:15:24,159 --> 00:15:27,399 scale rotate reflect shift images and 388 00:15:27,399 --> 00:15:30,580 more through matrix manipulation but 389 00:15:30,580 --> 00:15:32,740 things do get much more complicated like 390 00:15:32,740 --> 00:15:34,750 when you want to project a 3d image into 391 00:15:34,750 --> 00:15:37,809 a 2d plane we can use matrix map to map 392 00:15:37,809 --> 00:15:40,149 the 3d points and find where they would 393 00:15:40,149 --> 00:15:44,259 appear on the flat screen not going to 394 00:15:44,259 --> 00:15:45,730 go into much more detail in that but 395 00:15:45,730 --> 00:15:47,710 again computer graphics are another very 396 00:15:47,710 --> 00:15:51,279 useful application of matrices but for 397 00:15:51,279 --> 00:15:53,289 those wanting some real tangible results 398 00:15:53,289 --> 00:15:55,450 that come from matrix math let's look at 399 00:15:55,450 --> 00:15:59,320 networks and graph theory graphs can 400 00:15:59,320 --> 00:16:01,779 represent a lot of things people and who 401 00:16:01,779 --> 00:16:03,580 they're friends with connections on a 402 00:16:03,580 --> 00:16:06,009 dating app networks of cities and how 403 00:16:06,009 --> 00:16:08,019 they're connected websites and how they 404 00:16:08,019 --> 00:16:11,409 link to each other and so on with small 405 00:16:11,409 --> 00:16:13,240 networks it can be easy to intuitively 406 00:16:13,240 --> 00:16:16,000 understand what's going on like if I 407 00:16:16,000 --> 00:16:17,830 said here's a group of coworkers and 408 00:16:17,830 --> 00:16:19,779 connections represent mutual friendships 409 00:16:19,779 --> 00:16:21,879 it wouldn't be hard to see like this is 410 00:16:21,879 --> 00:16:24,250 the most popular person and this is the 411 00:16:24,250 --> 00:16:26,110 least popular with only one friend if 412 00:16:26,110 --> 00:16:28,149 you had to find how many mutual friends 413 00:16:28,149 --> 00:16:29,889 these two people have no big deal you 414 00:16:29,889 --> 00:16:33,009 can just count and see that's three but 415 00:16:33,009 --> 00:16:34,870 when the networks get more complex we 416 00:16:34,870 --> 00:16:36,279 need mathematical tools to help us 417 00:16:36,279 --> 00:16:39,220 identify key things this could be like 418 00:16:39,220 --> 00:16:40,480 which website should be ranked the 419 00:16:40,480 --> 00:16:42,009 highest on the web which I mentioned 420 00:16:42,009 --> 00:16:44,110 earlier it could be finding who is more 421 00:16:44,110 --> 00:16:45,909 likely to spread a disease and a college 422 00:16:45,909 --> 00:16:48,129 full of students for which people 423 00:16:48,129 --> 00:16:49,960 involved in the 9/11 terrorist attacks 424 00:16:49,960 --> 00:16:51,820 were most critical to the operation and 425 00:16:51,820 --> 00:16:53,820 should be prioritized by law enforcement 426 00:16:53,820 --> 00:16:56,620 yes they actually did this after 9/11 427 00:16:56,620 --> 00:16:57,519 which I have discussed in a previous 428 00:16:57,519 --> 00:17:00,190 video we need some mathematical 429 00:17:00,190 --> 00:17:02,080 techniques in these cases so we can find 430 00:17:02,080 --> 00:17:03,610 things that our eyes aren't always able 431 00:17:03,610 --> 00:17:05,849 to when the connections get this chaotic 432 00:17:05,849 --> 00:17:08,650 but let's do a matrices can reveal when 433 00:17:08,650 --> 00:17:11,680 it comes to dating apps imagine this app 434 00:17:11,680 --> 00:17:13,838 only has three men signed up that will 435 00:17:13,838 --> 00:17:16,569 label 1 through 3 and 3 women labeled 4 436 00:17:16,569 --> 00:17:17,959 through 6 and there 437 00:17:17,959 --> 00:17:21,859 together as shown these are mutual 438 00:17:21,859 --> 00:17:23,689 connections by the way like both people 439 00:17:23,689 --> 00:17:25,548 swiping right and you'll know for now 440 00:17:25,548 --> 00:17:28,970 there are no same-sex matches we see 441 00:17:28,970 --> 00:17:30,618 this first guys matched with all three 442 00:17:30,618 --> 00:17:32,929 women the second guy with two and the 443 00:17:32,929 --> 00:17:33,829 third with one 444 00:17:33,829 --> 00:17:36,319 now this graph provides a nice visual 445 00:17:36,319 --> 00:17:38,450 for this situation but what we can also 446 00:17:38,450 --> 00:17:40,878 do is make a table with six rows and six 447 00:17:40,878 --> 00:17:43,159 columns for the six people and analyze 448 00:17:43,159 --> 00:17:46,519 this instead we'll say if two people are 449 00:17:46,519 --> 00:17:48,319 matched like person one in person for 450 00:17:48,319 --> 00:17:50,569 our then in the square located and in 451 00:17:50,569 --> 00:17:54,339 column 1 and row four we will put a 1 452 00:17:54,339 --> 00:17:56,388 however since these are mutual 453 00:17:56,388 --> 00:17:58,730 connections we need to also include a 1 454 00:17:58,730 --> 00:18:02,359 in column 4 and Row 1 basically if one 455 00:18:02,359 --> 00:18:04,368 matches with 4 then of course 4 has 456 00:18:04,368 --> 00:18:05,749 matched with 1 so the data has to 457 00:18:05,749 --> 00:18:08,269 reflect that so that means yes the 458 00:18:08,269 --> 00:18:10,220 tables going to be symmetric about the 459 00:18:10,220 --> 00:18:13,368 diagonal then if two people are not 460 00:18:13,368 --> 00:18:16,038 connected like person 1 and 2 we'll put 461 00:18:16,038 --> 00:18:18,528 a 0 in that square in this case column 1 462 00:18:18,528 --> 00:18:20,599 in row 2 but of course we can't forget 463 00:18:20,599 --> 00:18:23,990 column 2 and row 1 since no one matches 464 00:18:23,990 --> 00:18:25,819 with themself the diagonal is going to 465 00:18:25,819 --> 00:18:28,249 be all zeros and then these would be the 466 00:18:28,249 --> 00:18:30,679 rest of the connections so if you want 467 00:18:30,679 --> 00:18:32,480 to know whether person 5 and 2 are 468 00:18:32,480 --> 00:18:34,970 connected by the table just go to column 469 00:18:34,970 --> 00:18:37,490 5 and Row 2 or vice versa and see if 470 00:18:37,490 --> 00:18:40,308 there's a one or a zero there from this 471 00:18:40,308 --> 00:18:42,829 there are obvious things we can see like 472 00:18:42,829 --> 00:18:44,509 for person 1 we can look down their 473 00:18:44,509 --> 00:18:46,970 column or across the row and find they 474 00:18:46,970 --> 00:18:48,710 have three matches in total because of 475 00:18:48,710 --> 00:18:50,720 the three ones but we're going to use 476 00:18:50,720 --> 00:18:52,249 some slightly more advanced math to 477 00:18:52,249 --> 00:18:54,319 analyze this graph so instead of 478 00:18:54,319 --> 00:18:56,179 considering this a table we're going to 479 00:18:56,179 --> 00:18:58,038 call it a matrix but it's taking away 480 00:18:58,038 --> 00:18:59,990 the gridlines but otherwise nothing has 481 00:18:59,990 --> 00:19:02,509 changed when it comes to graphs this is 482 00:19:02,509 --> 00:19:06,138 known as an adjacency matrix another way 483 00:19:06,138 --> 00:19:07,788 to interpret this though is that it 484 00:19:07,788 --> 00:19:09,950 tells us how many paths of length one 485 00:19:09,950 --> 00:19:13,909 exists between any two nodes oh and for 486 00:19:13,909 --> 00:19:15,710 the rest of this video when I say path I 487 00:19:15,710 --> 00:19:17,868 just mean any sequence of edges that 488 00:19:17,868 --> 00:19:19,730 joins a sequence of vertices basically 489 00:19:19,730 --> 00:19:22,429 just a walk those no graph theory may 490 00:19:22,429 --> 00:19:24,019 not like this because a path is usually 491 00:19:24,019 --> 00:19:26,179 more specific but I am being generic 492 00:19:26,179 --> 00:19:28,638 here okay so what's this really mean 493 00:19:28,638 --> 00:19:31,200 we'll look at column 6 and row 1 494 00:19:31,200 --> 00:19:33,390 we know this says that those two people 495 00:19:33,390 --> 00:19:35,009 are matched no big deal 496 00:19:35,009 --> 00:19:37,920 but it also means there is one path of 497 00:19:37,920 --> 00:19:40,380 length one that exists between them 498 00:19:40,380 --> 00:19:42,480 those are saying the same thing if we 499 00:19:42,480 --> 00:19:44,759 put a dot at person 1 and can only 500 00:19:44,759 --> 00:19:47,220 traverse one edge well there's one way 501 00:19:47,220 --> 00:19:49,980 to get to person 6 that's what this one 502 00:19:49,980 --> 00:19:52,589 represents on the other hand there are 503 00:19:52,589 --> 00:19:55,349 zero ways to get from person 1 to person 504 00:19:55,349 --> 00:19:58,619 2 in one edge if you start a person 1 505 00:19:58,619 --> 00:20:01,740 there are paths to person 2 but they all 506 00:20:01,740 --> 00:20:03,509 have a length of 2 which is not what we 507 00:20:03,509 --> 00:20:07,230 were looking for but now what if I want 508 00:20:07,230 --> 00:20:09,509 to see quickly how many mutual matches 509 00:20:09,509 --> 00:20:11,970 two people have well if we look at 510 00:20:11,970 --> 00:20:14,400 person 1 and 2 this isn't tough we see 511 00:20:14,400 --> 00:20:17,430 there are 2 mutual matches however this 512 00:20:17,430 --> 00:20:19,950 question of mutual matches is no 513 00:20:19,950 --> 00:20:22,500 different than asking how many paths of 514 00:20:22,500 --> 00:20:25,319 length 2 exists between person 1 and 515 00:20:25,319 --> 00:20:28,740 person 2 well we just saw that the 516 00:20:28,740 --> 00:20:30,779 answer is 2 as expected since it's the 517 00:20:30,779 --> 00:20:33,029 same question again if I start at 1 and 518 00:20:33,029 --> 00:20:36,720 go to 4 than 2 or 5 than 2 those two 519 00:20:36,720 --> 00:20:40,319 paths mean two mutual connections the 520 00:20:40,319 --> 00:20:41,819 cool thing though is that we can find 521 00:20:41,819 --> 00:20:43,500 how many of these paths of length 2 522 00:20:43,500 --> 00:20:46,049 exist between any two nodes by just 523 00:20:46,049 --> 00:20:47,940 multiplying the adjacency matrix by 524 00:20:47,940 --> 00:20:51,990 itself or squaring it we see for person 525 00:20:51,990 --> 00:20:54,390 1 and 2 there are 2 mutual connections 526 00:20:54,390 --> 00:20:56,609 so that checks out then if you look at 527 00:20:56,609 --> 00:20:58,619 the graph for person 2 and 3 they have 528 00:20:58,619 --> 00:21:01,140 no mutual connections and on the matrix 529 00:21:01,140 --> 00:21:03,359 this checks out as well from column 3 530 00:21:03,359 --> 00:21:07,410 and Row 2 having a 0 however person 1 531 00:21:07,410 --> 00:21:10,019 and 3 are both connected to 6 and no one 532 00:21:10,019 --> 00:21:11,640 else which we can also find in the 533 00:21:11,640 --> 00:21:15,660 matrix and you get the idea but now what 534 00:21:15,660 --> 00:21:17,970 would the diagonal mean well that's how 535 00:21:17,970 --> 00:21:20,009 many paths of length 2 it exists between 536 00:21:20,009 --> 00:21:21,569 a person and themselves 537 00:21:21,569 --> 00:21:24,779 aka how many matches they have think 538 00:21:24,779 --> 00:21:27,150 about it for person 1 if I start there 539 00:21:27,150 --> 00:21:30,450 to get back to 1 in two edges I can go 540 00:21:30,450 --> 00:21:34,619 into 4 than 1 5 than 1 or 6 and 1 3 541 00:21:34,619 --> 00:21:37,200 options for the 3 connections this is 542 00:21:37,200 --> 00:21:39,200 why we see a 3 there in the matrix 543 00:21:39,200 --> 00:21:42,119 person 2 that has two matches and it 544 00:21:42,119 --> 00:21:44,049 goes up 545 00:21:44,049 --> 00:21:46,519 then if we multiply the new matrix by 546 00:21:46,519 --> 00:21:48,680 the original the same as finding the 547 00:21:48,680 --> 00:21:51,109 original cubed we get all the paths of 548 00:21:51,109 --> 00:21:55,099 length 3 from one person to another see 549 00:21:55,099 --> 00:21:57,529 the original matrix to some power tells 550 00:21:57,529 --> 00:21:59,869 us how many paths of that length exist 551 00:21:59,869 --> 00:22:03,319 between any two notes now if we have a 552 00:22:03,319 --> 00:22:05,599 same-sex connection and link one to two 553 00:22:05,599 --> 00:22:07,880 let's say all we have to do is add a 1 554 00:22:07,880 --> 00:22:09,440 to the original matrix in the first 555 00:22:09,440 --> 00:22:11,559 column second row and vice versa 556 00:22:11,559 --> 00:22:13,579 see when implementing this in the 557 00:22:13,579 --> 00:22:15,079 software we just have to make small 558 00:22:15,079 --> 00:22:17,269 tweaks to the adjacency matrix and from 559 00:22:17,269 --> 00:22:19,220 there squaring or cubing it tells us a 560 00:22:19,220 --> 00:22:21,319 lot actually something I found 561 00:22:21,319 --> 00:22:23,329 interesting was what the matrix cube 562 00:22:23,329 --> 00:22:26,480 tells us specifically the diagonal for 563 00:22:26,480 --> 00:22:27,950 one it tells us how many paths of length 564 00:22:27,950 --> 00:22:29,539 three exists between a person and 565 00:22:29,539 --> 00:22:31,849 themselves but looking at the graph a 566 00:22:31,849 --> 00:22:34,700 length three path back to yourself tells 567 00:22:34,700 --> 00:22:37,940 us there's a triangle there I'm not 568 00:22:37,940 --> 00:22:39,680 going to explain this one in depth but 569 00:22:39,680 --> 00:22:41,390 if you sum the numbers along the 570 00:22:41,390 --> 00:22:43,309 diagonal also known as the trace of the 571 00:22:43,309 --> 00:22:46,069 matrix and divide by six that tells you 572 00:22:46,069 --> 00:22:48,589 how many triangles in total there are in 573 00:22:48,589 --> 00:22:51,019 the network I just found that to be a 574 00:22:51,019 --> 00:22:52,880 cool thing the matrix tells us which you 575 00:22:52,880 --> 00:22:56,269 wouldn't think about it first then on 576 00:22:56,269 --> 00:22:57,859 another topic something I haven't even 577 00:22:57,859 --> 00:22:59,630 mentioned yet is machine learning and 578 00:22:59,630 --> 00:23:01,940 neural networks which are coded with and 579 00:23:01,940 --> 00:23:04,029 manipulated by matrix math 580 00:23:04,029 --> 00:23:05,960 mathematically matrices are a huge 581 00:23:05,960 --> 00:23:07,700 aspect of what allows the machine to 582 00:23:07,700 --> 00:23:11,329 quote learn or in terms of security 583 00:23:11,329 --> 00:23:12,829 there's an example of an older kind of 584 00:23:12,829 --> 00:23:14,539 encryption method which is the hill 585 00:23:14,539 --> 00:23:17,450 cipher this cipher incorporates matrix 586 00:23:17,450 --> 00:23:19,309 operations in order to encrypt and 587 00:23:19,309 --> 00:23:21,559 decrypt messages although no it's not a 588 00:23:21,559 --> 00:23:24,319 modern encryption method and as much as 589 00:23:24,319 --> 00:23:26,029 I'd love to keep going into depth on 590 00:23:26,029 --> 00:23:27,710 different subjects this video is already 591 00:23:27,710 --> 00:23:29,930 quite long so hopefully this show just 592 00:23:29,930 --> 00:23:32,119 how powerful and impactful matrices are 593 00:23:32,119 --> 00:23:34,250 though but also regarding encryption and 594 00:23:34,250 --> 00:23:36,049 security I do want to thank dashlane for 595 00:23:36,049 --> 00:23:37,789 sponsoring this video a company that's 596 00:23:37,789 --> 00:23:39,680 dedicated to keeping you safe and secure 597 00:23:39,680 --> 00:23:42,740 on the internet - Lane is a password 598 00:23:42,740 --> 00:23:44,690 manager that well there's several things 599 00:23:44,690 --> 00:23:46,849 when it'll safely store all your 600 00:23:46,849 --> 00:23:49,160 passwords all in one place and sync them 601 00:23:49,160 --> 00:23:50,660 between devices so you never have to 602 00:23:50,660 --> 00:23:52,009 deal with resetting all those passwords 603 00:23:52,009 --> 00:23:53,660 you made months or years ago that you 604 00:23:53,660 --> 00:23:54,690 can't remember 605 00:23:54,690 --> 00:23:57,358 with this - lien will also autofill user 606 00:23:57,358 --> 00:23:59,159 names and passwords for any site you 607 00:23:59,159 --> 00:24:00,569 have stored in their vault making it 608 00:24:00,569 --> 00:24:01,950 just a little easier to navigate the 609 00:24:01,950 --> 00:24:04,169 internet something more important than 610 00:24:04,169 --> 00:24:05,848 remembering passwords though is having 611 00:24:05,848 --> 00:24:07,499 secure passwords and being safe from 612 00:24:07,499 --> 00:24:10,528 hackers which - lien takes care of we're 613 00:24:10,528 --> 00:24:11,669 totally shouldn't store the same 614 00:24:11,669 --> 00:24:13,648 password for different sites since being 615 00:24:13,648 --> 00:24:15,058 hacked in one place makes you vulnerable 616 00:24:15,058 --> 00:24:17,519 elsewhere now in dashlane that is solved 617 00:24:17,519 --> 00:24:18,628 as you have the option to have them 618 00:24:18,628 --> 00:24:21,210 auto-generate very secure passwords to 619 00:24:21,210 --> 00:24:22,710 be stored and used across all your 620 00:24:22,710 --> 00:24:25,558 devices you will have one single master 621 00:24:25,558 --> 00:24:26,819 password that you create when you sign 622 00:24:26,819 --> 00:24:28,589 up for dashlane and that is used to 623 00:24:28,589 --> 00:24:30,839 encrypt all others so even if someone 624 00:24:30,839 --> 00:24:32,489 were to gain access to internal servers 625 00:24:32,489 --> 00:24:35,398 they would just see gibberish you on the 626 00:24:35,398 --> 00:24:36,749 other hand can feel safe knowing you 627 00:24:36,749 --> 00:24:38,460 have different and very secure passwords 628 00:24:38,460 --> 00:24:40,558 for each site yet all you have to do is 629 00:24:40,558 --> 00:24:43,319 remember one single password plus four 630 00:24:43,319 --> 00:24:44,940 add security they'll notify you if any 631 00:24:44,940 --> 00:24:46,798 of your data has been compromised to 632 00:24:46,798 --> 00:24:48,210 give you a complete peace of mind as you 633 00:24:48,210 --> 00:24:49,138 browse the Internet 634 00:24:49,138 --> 00:24:51,388 pricing is already really cheap but if 635 00:24:51,388 --> 00:24:53,398 you sign up at the link below or go to - 636 00:24:53,398 --> 00:24:55,409 link comm slash major prep you'll get 637 00:24:55,409 --> 00:24:57,808 10% off your premium subscription plus 638 00:24:57,808 --> 00:24:59,638 there's a 30-day free trial so no risk 639 00:24:59,638 --> 00:25:01,798 and just giving it a try again links are 640 00:25:01,798 --> 00:25:03,179 below and with that I'm gonna end that 641 00:25:03,179 --> 00:25:05,489 video there you guys enjoyed be sure to 642 00:25:05,489 --> 00:25:06,808 LIKE and subscribe don't forget to 643 00:25:06,808 --> 00:25:08,638 follow me on Twitter hit the bell if 644 00:25:08,638 --> 00:25:09,898 you're not being notified and I'll see 645 00:25:09,898 --> 00:25:12,978 you all in the next video