Linear transformation of vectors

Question

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.

Answer 1

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$$P.A.P^{-1}=\begin{bmatrix}2 & 0 & 0 \\ 5 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}.$$