How can i form of the matrix representation in the following exercise?

Given the linear maps $f$ and $g$

$$f\left(\begin{bmatrix}\lambda_1\\ \lambda_2\\ \lambda_3\end{bmatrix}\right)=\begin{bmatrix}\lambda_3-\lambda_1\\\lambda_2-\lambda_1\end{bmatrix}\\ g\left(\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}\right)=\begin{bmatrix}\mu_2-\mu_1\\ -2\mu_1\\ \mu_1+\mu_2\end{bmatrix}$$

find the matrix repesentation of $g\circ f$.

  • $\begingroup$ Consider the first case, $f$. With what matrix do you need to multiply $$ \left[ \lambda_1, \quad \lambda_2, \quad \lambda_3 \right]^T $$ To obtain the matrix on the right side? Hint: It contains only $0$'s and $\pm 1$. $\endgroup$ – Matti P. Oct 23 '18 at 11:00

The matrix representation of $T:\Bbb R^n\to\Bbb R^m$ (in the canonical basis) is simply the matrix $$\begin{pmatrix}T(e_1)& T(e_2)&\cdots &T(e_n)\end{pmatrix}.$$

This can be easily computed in this instance.

  • $\begingroup$ Can you please suggest to me what does the cicle between the functions mean? $\endgroup$ – Róbert Kovács Oct 23 '18 at 11:09
  • 1
    $\begingroup$ I will not do that. $\endgroup$ – Saucy O'Path Oct 23 '18 at 11:10

$f$ is represented (in the standard bases) by $$M_f= \begin{bmatrix} -1 & 0 & 1\\ -1 & 1 & 0\end{bmatrix}$$

and $g$ by

$$M_g= \begin{bmatrix} -1 & 1\\ -2 & 0\\ 1 & 1 \end{bmatrix}$$

Now compute $M_g M_f$ as matrices


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