# Change of Basis (Transformation Matrix)

I am having some confusion regarding change of basis, and am unsure how to execute the following question:

If $$T\in \text{End}(V)$$ such that $$T(x_1)=2x_1+x_2$$ and $$T(x_2)=x_1$$, and $$y_1=4x_1+2x_2$$ and $$y_2=x_1-x_2$$, determine the matrix $$T$$ with respect to the basis $$\{x_1,x_2\}$$ and with respect to the new basis $$\{y_1,y_2\}$$. Furthermore, it is possible to find an invertible matrix $$P$$ such that $$P^{-1}AP=B$$, where $$A$$ is the matrix transformation with respect to the basis $$\{x_1,x_2\}$$ and $$B$$ is the matrix transformation with respect to the basis $$\{y_1,y_2\}$$.

For the first part, I know I need to find some matrix $$D=C^{-1}AC$$, such that $$A$$ is the transformation matrix with respect to the standard basis, and $$C$$ is the change of basis matrix, but I am unsure how to construct $$C$$ and thus $$C^{-1}$$. The transformation matrix for $$T$$ is:

$$A=\begin{bmatrix} 2 & 1\\ 1 & 0\\ \end{bmatrix}$$

I apologize regarding the simplicity of the question and would appreciate any help.

• What is the ‘standard’ basis for an abstract vector space? Jan 15, 2018 at 0:47
• Is it e_1,e_2,...,e_n? That is, (1,0,...,0),(0,1,...,0),...,(0,0,...,1)? Jan 15, 2018 at 0:50
• This makes sense for $\mathbf R^n$. But for an abstract vector space? All you know is it has dimension $2$. Jan 15, 2018 at 0:51

The matrix of $T$ in basis $(x_1,x_2)$ has for columns the coordinates of $T(x_1)$ and $T(x_2)$ in basis $(x_1,x_2)$. Thus $$A=\begin{bmatrix}2&1\\1&0\end{bmatrix}\quad\text{(no standard basis in this)}.$$ For the basis $(y_1,y_2)$, use the change of basis matrix from basis $(x_1,x_2)$ to basis $(y_1,y_2)$: its column vectoes are the coordinates of $y_1$ and $y_2$ in basis $(x_1,x_2)$, i.e. $$P=\begin{bmatrix}4&1\\2&-1\end{bmatrix}.$$ The change of basis formula asserts that the matrix of $T$ in the new basis is $$B=P^{-1}AP,\quad\text{so }\quad A=PBP^{-1}.$$ All that remains to do is calculating the inverse of $P$.