# Find the matrix $A'$ of $A:V \to W$ in the new bases $\{e_1', e_2'\}$ of $V$ and $\{f_1', f_2'\}$ of $W$.

Let $V$ and $W$ be 2-dimensional vector spaces, and let $A: V \to W$ be a linear map.

$e_1 = \begin{pmatrix}1\\1\end{pmatrix}, e_2 = \begin{pmatrix}0\\1\end{pmatrix}$ of $V$, and $f_1 = \begin{pmatrix}3\\4\end{pmatrix}, f_2 = \begin{pmatrix}2\\3\end{pmatrix}$ of $W$. The map $A$ has matrix $A = \begin{pmatrix}2 & 3\\ 3 & 5\end{pmatrix}$.

Let $e_1' = \begin{pmatrix} 2\\5 \end{pmatrix}, e_2' = \begin{pmatrix} 1\\3\end{pmatrix}$ be a new basis of $V$ and $f_1' = \begin{pmatrix} 2\\3\end{pmatrix}, f_2' = \begin{pmatrix} 1\\2\end{pmatrix}$ be a new basis of $W$.

Problem: Find the matrix $A'$ of $A:V \to W$ in the new bases $\{e_1', e_2'\}$ of $V$ and $\{f_1', f_2'\}$ of $W$.

I already did $[\operatorname{id}]_{e}^{e'} = \begin{pmatrix}2 & -1\\ -3 & 2\\\end{pmatrix}$ and $[\operatorname{id}]_f^{f'} = \begin{pmatrix} 2 & 1\\ -1 & 0\end{pmatrix}.$

After that, I think $$\left([\operatorname{id}]_e^{e'}\right)^{-1}A[\operatorname{id}]_f^{f'} = A'$$

I think that $e'\to e \to A \to f \to f'$. Is it correct? I don't know exactly how I multiple this matrices and What is the exact role of A(linear map).

• e'→ e → A → f → f ' – Daegun ko Apr 28 '17 at 4:23
• Please use MathJax. Formatting tips here. – Em. Apr 28 '17 at 4:53
• Please look here and here. – Itay4 Apr 28 '17 at 6:07
• I didn't understand your question, what don't you understand? – Ofek Gillon Apr 28 '17 at 13:24
• Actually I want to know about answer this problem – Daegun ko Apr 28 '17 at 14:37