Matrix of a map from the image
Task number: 2565
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 \( [f]_{KK}=\begin{pmatrix} 4 & 3 & 2\\ 2 & 1 & 3\\ 0 & 4 &1\\ \end{pmatrix} \)
