Let
where
If a basis is clear by context, then we may omit it and write
To show that the linear transformation is uniquely determined by these scalars, we will try to use the fact that elements in each vector space are already uniquely determined by their own constants and go from there.
Let
Now given a generic
The lets us conclude that
To see what the matrix is, we can try to figure out what it's columns are.
Recall that we can extract the column of a matrix using the column extraction method discussed earlier
To be successful at that we need to generate a unit column matrix. We can do so by plugging in
Thus
First of all note that these two vectors must be linearly independent, if they were dependent, then the by comparing the first component of these two vectors we would see that the second is twice the first, whereas comparing the second component would tell us that the second vector is
Since the standard basis for
Now we can use the casting out method to determine which of the column vectors are linearly dependent on the other vectors, we start by putting all the column vectors as the columns of a matrix and then row reduce, which yields
By the casting out method, we know that since the fourth and fifth columns are non-pivot columns and therefore the vectors