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It's been a few weeks since the subject was covered in my Linear Algebra class, and unfortunately linear transformations are my weak spot, so could anyone explain the steps to solve this problem?

Find the matrix $A$ of the linear transformation $T(f(t)) = 3f'(t)+7f(t)$ from $P_2$ to $P_2$ with respect to the standard basis for $P_2$, $\{1,t,t^2\}$.

The resulting answer should be a $3 \times 3$ matrix, but I'm unsure of where to start when it comes to solving this problem.

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    The first column of the matrix is the vector $T(1)$, the second column is the vector $T(t)$, and the third column is the vector $T(t^2)$. Try it out. All these vectors should be written with their coordinates in terms of the basis $\{1, t, t^2\}$, but luckily this is a easy basis to work with in $P_2$.2012-12-08
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    @HenryT.Horton as in evaluate the equation for each?2012-12-08
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    Yes. For example, $T(3 + 2t) = 3(3 + 2t)' + 7(3 + 2t) = 3 \cdot 2 + 21 + 14t = 27 + 14t + 0t^2$, so $T(3 + 2t)$ would correspond to the vector $$\begin{pmatrix} 27 \\ 14 \\ 0 \end{pmatrix}.$$2012-12-08
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    Ahh! So then I express my resulting answer after evaluation in terms of being a combination of the three parts of my basis? That makes perfect sense...Thanks alot @HenryT.Horton2012-12-08
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    Yes. In this case there isn't any extra work to be done, since the basis here is so easy to write things as a linear combination of. In general, you would have to solve a system of linear equations to find out what the components of the resulting vector with respect to the given basis is.2012-12-08
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    Aside: linear algebra is the study of linear transformations. If linear transformations are your weak spot, you should be seriously worried.2012-12-08
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    @ChrisEagle For the scope of my class, linear transformations are the only thing i struggle with, I'm unfortunately stuck with a teacher who's teaching style makes it very hard to learn in class. I've got a strong grasp of RREF and Eigenvalues, which pretty much runs the range of what we were able to cover in the semester. I understand the concept of linear transformations, i just have trouble grasping the method of solving some problems.2012-12-08

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NOTE Given a finite dimensional vector space $\Bbb V$ and a basis $B=\{v_1,\dots,v_n\}$ of $\Bbb V$, the coordinates of $v$ in base $B$ are the unique $n$ scalars $\{a_1,\dots,a_n\}$ such that $v=\sum_{k=1}^n a_kv_k$, and we note this by writing $(v)_B=(a_1,\dots,a_n)$.


All you need is to find what $T$ maps the basis elements to. Why? Because any vector in $P_2$ can be written as a linear combination of $1$, $t$ and $t^2$, whence if you know what $T(1)$, $T(t)$ and $T(t^2)$ are, you will find what any $T(a_0 +a_1 t +a_2 t^2)=a_0T(1)+a_1 T(t)+a_2 T(t^2)$ is. So, let us call $B=\{1,t,t^2\}$. Then

$$T(1)=3\cdot 0 +7\cdot 1=7=(7,0,0)$$

$$T(t)=3\cdot 1 +7\cdot t=(3,7,0)$$

$$T(t^2)=6\cdot t +7\cdot t^2=(0,6,7)$$

(Here I'm abusing the notation a bit. Formally, we should enclose the two first terms of the equations with $(-)_B$ )

Now note that our transformation matrix simply takes a vector in coordinates of base $B$, and maps it to another vector in coordinates of base $B$. Thus, if $|T|_{B,B}$ is our matrix from base $B$ to base $B$, we must have

$$|T|_{B,B} (P)_B=(T(P))_B$$

where we wrote $P=P(t)$ to avoid too much parenthesis.

But let's observe that if $(P)_B=(a_0,a_1,a_2)$ then $a_0T(1)+a_1 T(t)+a_2 T(t^2)=a_0(7,0,0)+a_1 (3,7,0)+a_2(0,6,7)$ is the matrix product

$$\left(\begin{matrix}7&3&0\\0&7&6\\0&0&7\end{matrix}\right)\left(\begin{matrix}a_0 \\a_1 \\a_2 \end{matrix}\right)$$

And $|T|_{B,B}=\left(\begin{matrix}7&3&0\\0&7&6\\0&0&7\end{matrix}\right)$ is precisely the matrix we're after. It has the property that for each vector of $P_2$

$$|T|_{B, B}(P)_B=(T(P))_B$$

(well, actually

$$(|T|_{B,B} (P)_B^t)^t=(T(P))_B$$

but that looks just clumsy, doesn't it?)