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

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.

• 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? Feb 22 '21 at 11:55

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}

• Thank you very much, it was very helpful, do you have any tip for b)?
– Erza
Feb 22 '21 at 14:35
• @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. Feb 22 '21 at 14:37
• 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?
– Erza
Feb 22 '21 at 14:47
• @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? Feb 22 '21 at 14:53
• 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?
– Erza
Feb 22 '21 at 15:01