Dr, Robert van de Geijn In the last unit, we saw that a linear
transformation can be represented as a matrix and that applying a linear
transformation to a vector is equivalent to performing a matrix
vector multiplication.
What we're going to see in this unit is that a vector function is a linear
transformation if and only if it can be represented by a matrix and is
equivalent to a matrix vector multiplication.
We then move on and show that we can exploit that to more definitively be
able to say whether a vector function is or is not a linear transformation.
So here is what I just said as a theorem.
If we have a vector function from Rn to Rm defined by l of x is equal to A
times x, then l is a linear transformation.
We're going to leave the proof of this as an exercise.
But what this shows is that l is a linear transformation if and only if
it can be represented by a matrix A and l of x is equal to A times x.
Now let's see how we can exploit that.
Last week, we asked the question, is the function that transforms the
vector chi 0, chi 1 into the vector chi 0 plus chi 1 for the first
component, and chi 0 for the second component a linear transformation.
Here's an alternative proof.
What we're going to do is we're going to compute the possible matrix that
represents f.
And then we're going to see whether matrix vector multiplication with
matrix gives us the exact same result as applying the function f.
If the answer is yes, then f is a linear transformation.
If the answer is no, then it is not.
So how do we compute the possible matrix that represents f?
Well, we evaluate a first unit basis vector.
And if we take 1, 0 and plug it in here, we get 1 plus 0, 1, and we get
this vector right here.
And if we plug-in the second unit basis vector, then we plug-in 0 and 1
in for the appropriate quantities in the function, we get
this, and we get that.
Now, we know that the matrix that represents a linear transformation f
if it were a linear transformation now has, as its first
column, this right here.
Why?
Because it's the first unit basis vector transformed by f.
And this is its second column.
So we create that matrix.
Now if we apply a times x, then we apply this matrix times this vector.
And if we multiply that out, we get this expression right here.
But notice that that expression is exactly the function
that we started with.
And therefore, we conclude that f of x is equal to A times x.
And therefore, it must be a linear transformation.
Let's look at another example.
Now we're going to show that this function here is not a linear
transformation.
We already did that by coming up with vectors that did not satisfy the
definition of a linear transformation.
But let's look at it a little bit differently.
Again, we come up with the possible matrix that represents f.
How did we do that?
We evaluate f for the first unit basis vector, that gives us this, we
evaluate f for the second unit basis vector, that gives us that, we take
this as the first column, and this as the second column, of the matrix.
That gives us this matrix A right here.
And then we see whether A times x is indeed the same as
evaluating the function.
A times x means multiplying this matrix right here.
This gives is this times that plus this times that for the first
component and this times that plus this times that
for the second component.
And what we notice is that that's not the same as our function.
Therefore, our function cannot be a linear transformation.
Otherwise, there would have been this matrix, and the matrix vector
multiplication would have given us the same result.
By understanding that a vector function is a linear transformation if
and only if it has the same action as a matrix vector multiply with the
appropriate matrix, we now have a foolproof way of checking whether a
function is a linear transformation.
You compute the possible matrix.
You check if the matrix vector multiply gives us the same result as
the function, and we're done.