# Find the matrix #A# for the linear transformation #T# relative to the bases #B = {1,x,x^2}# and #B' = {1,x,x^2,x^3}# such that #T(vecx) = Avecx#?

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Let #T:P^2 |-> P^3# given by #4x*vecp# be a linear transformation. (I'm guessing #vecp# is a vector of polynomial terms?)

I couldn't follow my professor in class when he did linear transformations involving nonstandard bases, because the notation was confusing... So I'm really not sure how to do this.

Let

I couldn't follow my professor in class when he did linear transformations involving nonstandard bases, because the notation was confusing... So I'm really not sure how to do this.

##### 1 Answer

#### Explanation:

The idea behind the notation of using *are* polynomials. Just as in

You can have a nonstandard basis in

Similarly, you can have nonstandard bases in *that basis*.

While this problem uses the standard bases for

For the problem itself, when we wish to find the matrix representation of a given transformation, all we need to do is see how the transformation acts on each member of the original basis and put that in terms of the target basis. The resulting vectors will be the column vectors of the matrix.

First, we see how

Then we use those as the column vectors for our transformation matrix:

Let's check to see if it works.

Given a polynomial