# Find matrix of linear transformation in a new basis

Linear transformation $\varphi$ in the basis $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3,\mathbf{e}_4$

Has matrix:

$$A = \begin{pmatrix} 1 & 2 &0 & 1 \\ 3 & 0 & -1 & 2 \\ 2 & 5 & 3 & 1 \\ 1 & 2 & 1 & 3\end{pmatrix}$$

Find matrix of this linear transformation in the basis of $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3,\mathbf{e}_4$

(I do not know why the same basis is given, but anyway I had different answer.)

What I know now:

It is well known that matrix in a new basis can be found by the formula:

$$A' = T^{-1} \cdot A \cdot T$$

I am a bit confused what is $T$ here?

I wrote $T$ like this:

$$T = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 &0 & 0 & 1\end{pmatrix}$$

But it does not seem to be correct,because as a result I got wrong answer.

Should I just put new basis as $T$ each time, or write $T=(\textbf{old basis}|\textbf{new basis})$ and find $T$ as the Transformation matrix using gaussian elimination?

• The main question in my head is how $T$ should be handled? should I just put new basis into $T$ or find $T$ as a Transformation matrix like: $T = (\textbf{old basis} | \textbf{new basis})$ and apply gaussian elimination? Jun 4, 2017 at 18:02
• T should indeed be the transformation matrix of the old to the new basis. So the element $T_{ij}$ should be the expansion coefficient of the new basis vector $e_i'$ with respect to the old vector $e_j$. Jun 4, 2017 at 18:29