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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?

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1 Answer 1

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You gave the right formula yourself. And if they ask to calculate it in the same basis then the transformation matrix is indeed the identity matrix.

So there two possibilities. Or you made a mistake during your calculation or they made a mistake with the new basis.

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  • $\begingroup$ 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? $\endgroup$
    – M.Mass
    Jun 4, 2017 at 18:02
  • $\begingroup$ 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 $e_j$. $\endgroup$
    – NDewolf
    Jun 4, 2017 at 18:29

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