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First, sorry for the unclear title but I can't think of any better one. Here I my homework and I don't under stand it

Let $\varepsilon = \{e_1,e_2,e_3\}$ be the standard matrix for $R^3$, let $\beta=\{b_1,b_2,b_3\}$ be a basis for a vector space V and let $T:R^3 \to V$ be a linear transformation with properties that: $$T(x_1,x_2,x_3)=(2x_3 - x_2)b_1 - (2x_2)b_2 + (x_1+3x_3)b_3$$

a. compute $T(e_1),T(e_2),T(e_3)$

b. compute $[T(e_1)]_\beta,[T(e_2)]_\beta,[T(e_3)]_\beta$

c. find the matrix T relative to $\varepsilon$ and $\beta$

Honestly, I don't under stand what $T(x_1,x_2,x_3)$ means. Why there are commas inside the brackets. I've never encountered that kind of linear transformation before. Can anyone explain this kind of linear transformation for me. Thank a lot

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1 Answer 1

up vote 1 down vote accepted

$T(x_1,x_2,x_3)$ is the image of a point $(x_1,x_2,x_3) \in \mathbb{R}^3$. For instance, $T(1,0,0) = T(e_1) = b_3$

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so if $x=\begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix}$ then T(x) is the same with $T(x_1,x_2,x_3)$? –  aukxn Feb 25 at 4:00
    
correct......... –  user130448 Feb 25 at 4:04
    
thank, that was new to me. –  aukxn Feb 25 at 4:44

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