# Linear transformation of vectors

We have vectors:

$$a=\begin{bmatrix} 1\\ 2\\ 0 \end{bmatrix}b=\begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix} c=\begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$$

The linear transformation $$A:R^3 \rightarrow R^3$$ is :

$$A a=a+b$$ $$Ab=2a$$ $$Ac=c$$

We need to find the matrix for basis $$B=\{a,b,c\}$$ and the standard basis of $$R^3$$

I tried:

I would put the vectors of linear transformation in rows $$A \begin{bmatrix} a\\ b\\ c \end{bmatrix}=\begin{bmatrix} 2 & 3 &0\\ 2 & 4 &0\\ 0& 0& 1 \end{bmatrix}$$ And then I would do matrix multiplication:

$$\begin{bmatrix} 2 & 3 &0\\ 2 & 4 &0\\ 0& 0& 1 \end{bmatrix}\begin{bmatrix} 2 \\ 3\\ 0 \end{bmatrix}=...$$

I would get the wrong answer.

It follows from the definition of $$A$$ that is matrix with respect to $$B$$ is$$\begin{bmatrix}1&2&0\\1&0&0\\0&0&1\end{bmatrix}.$$This is independent of the definitions of $$a$$, $$b$$, and $$c$$.
Now, if$$P=\begin{bmatrix}1&1&0\\2&1&0\\0&0&1\end{bmatrix},$$(the columns of $$P$$ are the vectors of $$B$$) then the matrix of $$A$$ with respect to the standard basis is$$P.A.P^{-1}=\begin{bmatrix}2 & 0 & 0 \\ 5 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}.$$