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Suppose $T:\mathbb{R}^3\to\mathbb{R}^2$ is a linear transformation with: $$T(e_1)=\begin{bmatrix}3\\1\end{bmatrix},\qquad T(e_2)=\begin{bmatrix}4\\1\end{bmatrix},\qquad T(e_3)=\begin{bmatrix}5\\9\end{bmatrix}.$$

a) Determine the matrix which represents $T$.

b) Compute $T\begin{bmatrix}2\\3\\-1\end{bmatrix}$ (Is the answer for this $\begin{bmatrix}13\\4\end{bmatrix}$?)

c) Write the range of $T$ as the span of a set of linearly independent vectors.

d) Is the vector $\begin{bmatrix}\pi\\\sqrt{2}\end{bmatrix}$ in the range of $T$?

Please give explanations and show steps with answers. I really want to learn how to do this... Thank you...

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What about a)? Any idea on how to do that? – 1015 Mar 11 '13 at 3:00
Walter, just changing your name isn't going to change the fact that you need to know the definitions of the span of a set of vectors, what linearly independent vectors. Your earlier post indicates you haven't any grasp of those definitions, and you need to understand those concepts in order to do your work. – amWhy Mar 11 '13 at 3:00
@julien: See this post – amWhy Mar 11 '13 at 3:01
I changed the name because it some how didn't let me post the question... That's not important though. I know what they mean. There were numerous questions given, and these were the only ones I couldn't solve... If I knew how to solve them, I wouldn't have came here.... – Walter Mar 11 '13 at 3:02
@Walter: Please look at the revisions I made to see how to format your posts better in the future. You can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. – Zev Chonoles Mar 11 '13 at 3:03

Since $T:\mathbb{R}^3\to\mathbb{R}^2$, then you are looking for a 2 by 3 matrix. If you don't know already,

$e_1=\begin{bmatrix}1\\0\\0\end{bmatrix},\qquad e_2=\begin{bmatrix}0\\1\\0\end{bmatrix},\qquad e_3=\begin{bmatrix}0\\0\\1\end{bmatrix}$

Since $T(e_1)=\begin{bmatrix}3\\1\end{bmatrix}$, and $T(e_1)=T\begin{bmatrix}1\\0\\0\end{bmatrix}=$ the first column of $T$ (try multiplying out to see why), then you know that the first column of $T$ is $\begin{bmatrix}3\\1\end{bmatrix}$

Then repeat this reasoning for the other columns to get $T$.

Once you know what $T$ is, then use the definition of matrix-vector multiplication to answer part b. (If I am not mistaken, $\begin{bmatrix}13\\4\end{bmatrix}$ is close to the answer.)

For part c, think about the definition of range, and the relationship between vectors in the range and $T$. It should consist of vectors in $\mathbb{R}^2$, by looking at $T$. Thus, you need at most 2 vectors (any more than that, and the set won't be linearly independent anymore).

Part d is true if you can write $\begin{bmatrix}\pi\\\sqrt{2}\end{bmatrix}$ as a linear combination of the two linearly independent vectors in part c, and false otherwise.

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Thanks you so much for the explanations. Ok, so I tried to work it out. So is a) Is the matrix which represents T, [3,1],[4,1],[5,9]? As for c), are [3,1],[4,1],[5,9] the range or codomain? They always confuse me... and why at most 2 vectors? And for part d, from what do I find the two linearly independent vectors that you are referring to to write [π2√]? – Walter Mar 11 '13 at 3:32

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