# How to give matrix representation of linear maps

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 $$g\circ f$$.

• 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 $\pm 1$. – 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.

• Can you please suggest to me what does the cicle between the functions mean? – Róbert Kovács Oct 23 '18 at 11:09
• I will not do that. – 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