I am having trouble with this problem. I have to find the matrix representation of a linear transformation. The example in my book got me my answer below but I do not feel that it is right/sufficient. Can someone explain matrix representation of a linear transformation?
Given $P_2(x)$ and $P_3(x)$ and the linear transformation: $L:P_2(x)\rightarrow P_3(x)$ defined by $L(p(x)) = \displaystyle \int p(x)dx$. Find the matrix representation $A$ of the linear transformation $L$. Then find the rank of $A$ and the null space of $A$.
Here is what I have:
$$A = \begin{bmatrix}0&1&0\\ 0&0&2\\ 0&0&0\end{bmatrix}$$
$R(A)$ = 2
$N(A)$ = 1
Best Answer
choose a basis for the polynomial spaces, say $\{1,x,x^2\}$ and $\{1,x,x^2,x^3\}$. then integration $\int_0^xp(t)dt$ takes the basis for $P_2$ to $x,x^2/2,x^3/3$. in terms of vectors $$ (1,0,0)\mapsto(0,1,0,0), (0,1,0)\mapsto(0,0,1/2,0), (0,0,1)\mapsto(0,0,0,1/3) $$ so you get the matrix (wrt these bases) $$ \left( \begin{array}{ccc} 0&0&0\\ 1&0&0\\ 0&1/2&0\\ 0&0&1/3\\ \end{array} \right) $$