# Thread: Find the Matrix for T Relative to the bases {1, t, t^2} and {1, t, t^2, t^3}.

1. ## Find the Matrix for T Relative to the bases {1, t, t^2} and {1, t, t^2, t^3}.

I've done part a and b. I'm stuck on c and don't know how to go about it. Can somebody help please?

2. ## Re: Find the Matrix for T Relative to the bases {1, t, t^2} and {1, t, t^2, t^3}.

Notice that the matrix product
$\displaystyle \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}= \begin{bmatrix}a \\ d \\ g\end{bmatrix}$

That is, the matrix times that vector, with 1 in the first place and all others 0, is the first column of the matrix.
You should be able to see that the matrix times the vector, with one in thesecond place, is the second column of the matrix.
Etc.

So apply the linear transformation to each vector of the first basis, in turn, writing the result in terms of the second basis. The coefficients will give each column, in turn, of the matrix.

For example, the first basis "vector" is p(t)= 1. The given linear transformation maps each polynomial into that same polynomial times t+ 3. In particular 1 is mapped into t+ 3. The second basis is $\displaystyle \{1, t, t^2, t^3\}$. As a linear combination of those (in that order), $\displaystyle t+ 3= 3(1)+ 1(t)+ 0(t^2)+ 0(t^3)$. Therefore the first column in the matrix representation is $\displaystyle \begin{bmatrix}3 \\ 1 \\ 0 \\ 0\end{bmatrix}$.