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The Question

With respect to the bases,

$$a_1 =\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},\ a_2 =\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix},\ a_3 =\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \text{of}\ \mathbb{R}^3$$

$$b_1 =\begin{pmatrix} -1 \\ 2 \end{pmatrix},\ b_2 =\begin{pmatrix} 1 \\ -1 \end{pmatrix} \text{of}\ \mathbb{R}^2$$

the linear mapping of $L:\mathbb{R}^3 \rightarrow \mathbb{R}^2$ is represented by the matrix:

$$A =\begin{pmatrix} 1 & 0 & 3 \\ -2 & 2 & 0 \end{pmatrix}.$$

a) Calculate the representation matrix of $L$ with respect to the standard bases of $\mathbb{R}^3$ and $\mathbb{R}^2.$

b) Give a basis of $c_1, c_2, c_3$ of $\mathbb{R}^3$ and a basis $d_1, d_2$ of $\mathbb{R}^2$ such that $L$ is represented by the matrix

$$A_0 =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$$

with respect to these bases.

The Understanding

Can someone help me solve this task or at least explain me how it should be done. I don't understand this very well.

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  • $\begingroup$ Please use the basic tutorial and quick reference guide and use mathjax for future questions, as images cannot be searched on MSE. Also, what previous attempts have you done? $\endgroup$
    – Laufen
    Feb 22 '21 at 11:55
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Notice that we have a linear transformation from domain $\mathbb{R}^3$ to the co-domain $\mathbb{R}^2$.

(a) It is given that linear transformation L is represented by:

\begin{align} A =\begin{pmatrix} 1 & 0 & 3 \\ -2 & 2 & 0 \end{pmatrix} \end{align}

Now we want to find the representation matrix for the linear transformation L with respect to the bases standard basis $\beta = \{(1,0,0),(0,1,0),(0,0,1)\}$ and $\gamma = \{(1,0),(0,1)\}$.

\begin{align} T((1,1,1)) = A\begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix} =\begin{pmatrix} 1 & 0 & 3 \\ -2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix} = \begin{pmatrix} 4 \\ 0 \\ \end{pmatrix} \end{align}

\begin{align} T((0,1,2)) = A\begin{pmatrix} 0 \\ 1 \\ 2 \\ \end{pmatrix} =\begin{pmatrix} 1 & 0 & 3 \\ -2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 2 \\ \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \\ \end{pmatrix} \end{align}

\begin{align} T((1,0,0)) = A\begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} =\begin{pmatrix} 1 & 0 & 3 \\ -2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \\ \end{pmatrix} \end{align}

Now expressing these vectors as linear combinations of b1,b2,b3:

T(a1) = 4b1 + 8b2 (verify)

T(a2) = 8b1 + 14b2 (verify)

T(a3) = -1b1 + 0b2 (verify)

So the desired matrix: \begin{align} \begin{pmatrix} 4 & 8 & -1 \\ 8 & 14 & 0 \end{pmatrix} \end{align}

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  • $\begingroup$ Thank you very much, it was very helpful, do you have any tip for b)? $\endgroup$
    – Erza
    Feb 22 '21 at 14:35
  • $\begingroup$ @Erza So I take that you are convinced with the solution of part a? about part b, I need to check something, then I will add the solution to the post. $\endgroup$ Feb 22 '21 at 14:37
  • $\begingroup$ Ok, thank you very much. For part a) I´ve done the task until there where you wrote (verify) in the same way as you did, but in the end I got the same Matrix as was given and I wasn´t sure about my solution, because I didn´t know that I should get the Matrix from these numbers (4 8), (8 14), (-1 0), so are you very sure it should be done in this way? $\endgroup$
    – Erza
    Feb 22 '21 at 14:47
  • $\begingroup$ @Erza, yes, I'm positive. The procedure is to compute the a's using the linear transformation (because they are in the domain of L) after that, you need to express them as a linear combination of the b's. So after doing that, you will get the matrix that represents these changes. Clear? $\endgroup$ Feb 22 '21 at 14:53
  • $\begingroup$ Yes, clear. Thanks. And for example, if it´s a Matrix given with respect to Standardbases and we have to calculate the representation matrix of L with respect to other bases, it should be done in the same way right? $\endgroup$
    – Erza
    Feb 22 '21 at 15:01

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