## Matrix of a map from the image

Determine the matrix of $$f:\mathbb Z_5^3\to \mathbb Z_5^3$$ w.r.t. the canonical basis $$K$$ (the same in both spaces). The mapping $$f$$ maps vectors $$u_1=(2{,}4,1)^T$$, $$u_2=(2{,}3,4)^T$$ and $$u_3=(3{,}0,1)^T$$ onto vectors $$f(u_1)=(2{,}1,2)^T$$, $$f(u_2)=(0{,}4,1)^T$$ and $$f(u_3)=(4{,}4,1)^T$$.

• #### Hint

Use $$[f(u)]_K=[f]_{KK}[u]_K$$.

• #### Resolution

We solve the matrix equation $$\begin{pmatrix} 2 & 0 & 4\\ 1 & 4 & 4\\ 2 & 1 &1\\ \end{pmatrix} =[f]_{KK} \begin{pmatrix} 2 & 2 & 3\\ 4 & 3 & 0\\ 1 & 4 &1\\ \end{pmatrix}$$.

We multiply this from the right by $$\begin{pmatrix} 2 & 2 & 3\\ 4 & 3 & 0\\ 1 & 4 &1\\ \end{pmatrix}^{-1}= \begin{pmatrix} 4 & 0 & 3\\ 3 & 2 & 1\\ 4 & 2 &4\\ \end{pmatrix}$$.

• #### Result

The matrix of the map is $$[f]_{KK}=\begin{pmatrix} 4 & 3 & 2\\ 2 & 1 & 3\\ 0 & 4 &1\\ \end{pmatrix}$$