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Let $P \in \mathbb{R}^{3 \times 3}$ be an invertible matrix. I need to determine whether the transformation $T : \mathbb{R}^{3 \times 3} \to \mathbb{R}^{3 \times 3}$ defined by $T(A) = P^{-1}AP$ is a linear transformation or not. Thanks a lot

Edition: Thanks everybody, now how can I know if it is one to one, and if it is onto? For one to one I'm trying to take two matrices $A_1$ and $A_2$, assume that $T(A_1) = T(A_2)$, and discover whether $A_1 = A_2$, but arrived to a deadlock. Any tips?

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  • $\begingroup$ What do you need to check and what have you already checked? $\endgroup$ – DonAntonio Mar 29 '16 at 20:32
  • $\begingroup$ A linear transform is one to one iff its kernel is the null vector. To check if it is onto, let $M$ be a matrix. It is the matrix of a linear transform $u$ on $\mathbb R^3$ relative to the usual basis. Let $\mathcal B$ be the basis of that vector space obtained from the columns of $P$, which is invertible. Express the matrix of $u$ in that basis. $\endgroup$ – Groovy. Mar 30 '16 at 10:58
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Hint:

The key facts are:

1) The matrix multiplication is left and right distributive with respect to the addition, so: $$ A(B+C)=AB+AC \qquad (A+B)C=AC+BC \qquad \forall A,B,C $$

2) For matrices with entries from a field, matrix multiplication is compatible and commute with scalar multiplication:

$$ A(\lambda B)=\lambda (AB)=(\lambda A) B $$

Now use these to prove linearity as suggested on the other answers.

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  • $\begingroup$ Thanks a lot, it was useful.... $\endgroup$ – A-H Mar 29 '16 at 21:09
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To show something is a linear transformation you must show that $T(ax + y) = aT(x) + T(y)$. If you try this you will find your answer.

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$\textbf{Hint:}$
Lineair means in this case: $$ T(A+\lambda B) = T(A)+\lambda T(B) $$ for any $\lambda \in \mathbb{R}$. What happens when you plug this in?

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