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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

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    @jme$n$degan The following link may be useful. http://math.stackexchange.com/questions/64907/correspondences-between-linear-transformations-and-matrices/64911#649112011-12-05

2 Answers 2

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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) $

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    the nullspace is 02011-12-08
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Adding to @yoyo answer.

Mechanical way is:

$x \mapsto y$
0: $(1,0,0)\mapsto(0,1,0,0)$
1: $(0,1,0)\mapsto(0,0,1/2,0)$
2: $(0,0,1)\mapsto(0,0,0,1/3)$
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n: . . .

trannsformation required = $\displaystyle \sum\limits_0^n y^Tx$

Every $y^Tx$ will give you $4 \times 3$ matrix and you add them up.