Problem about matrix of linear transformation

Here is my problem:

Let $$V$$ be the vector space of polynomials of degree up to $$2$$.

and $$T:V\rightarrow V$$ be a linear transformation defined by the type:

$$T(p(x))=p(2x+1)$$

Find the matrix form of this linear transformation. The base to find the matrix is $$B=\{1,x,x^2\}$$

I have seen in the past other exercises about finding the matrix of a linear transformation but they were all simpler and I have no idea how to proceed with this one. Any help will be appreciated.

• To find a matrix, you need a basis – J. W. Tanner Sep 11 at 10:02
• @J.W.Tanner Yes you are right. The exercise said to use the "usual basis" $B=\{1,x,x^2\}$ – michail vazaios Sep 11 at 10:05
• How did you solve other similar exercises? Did you try doing precisely the same thing for this problem? Where did you get stuck? – HerrWarum Sep 11 at 10:06
• @HerrWarum I've solved similar exercises where the type of the linear transformation was given explicitly. It was given like $T((x,y)^T)=(2x+y,x-y)^T$. Where I got stuck is that I can't the type(or such a type for this transformation). – michail vazaios Sep 11 at 10:12
• To work out what this transformation does to the basis vectors, simply substitute $2x+1$ for $x$ in them. That’s what the transformation does. – amd Sep 11 at 17:17

This means $$T(ax^2+bx+c)=(4ax^2+(4a+2b)x+a+b+c)$$. Taking the bais as $$(x^2,x,1)$$, we can write $$T \begin{pmatrix} a \\ b \\c \end{pmatrix}= \begin{pmatrix} 4a \\ 4a+2b \\a+b+c \end{pmatrix} \Rightarrow \begin{pmatrix} 4 & 0 & 0 \\ 4 & 2 & 0 \\ 1 & 1 & 1 \end{pmatrix}\begin{pmatrix} a \\ b \\c \end{pmatrix}.$$ So the required $$T$$ matrix is $$T= \begin{pmatrix} 4 & 0 & 0 \\ 4 & 2 & 0 \\ 1 & 1 & 1 \end{pmatrix}.$$