Matrix Multiplication || Linear Transformation This seems to be trivial question but unfortunately I can't figure it out:
Here $B,V$ and $U$ are matrices:
Do relation $B(V + U)B^{-1} = BVB^{-1} + BUB^{-1}$ hold true?
If yes than which matrix property is exploited? 
P.S. : If yes than $T(A) = BAB^{-1}$ will be a linear transformation. 
 A: That property is called linearity
All matrix multiplications are a linear transformation.
What you have written has a special name: Similarity Transformation
https://en.wikipedia.org/wiki/Matrix_similarity
http://mathworld.wolfram.com/SimilarityTransformation.html
A: Note, that matrix multiplication is, among other things, distributive, that is:
$$(A+B)C = AB+BC \text{ and } A(B+C) = AB+AC$$
for all matrices $A,B,C$, s.t. the above expressions are defined.
We can even show: For all $n\in \mathbb{N}$, the set of $n\times n$-Matrices equipped with matrix addition and matrix multiplication forms a ring, hence (in this case) the following properties hold:


*

*$(A+B)+C = A+(B+C)$

*$A+B = B+A$

*$A + 0 = A$

*$A + (-A) = 0$

*$(AB)C = A(BC)$

*$AI = IA = A$

*$(A+B)C = AB+BC$

*$A(B+C) = AB+AC$


for all $n\times n$-matrices $A,B,C$; where $0$ is the zero matrix (containing only zeros) and $I$ is the identity matrix. This situation is similar to, what we know from integers, except matrix multiplication is not commutative.
The properties of multiplication even hold, when the matrices are not all of the same dimension, if the expressions are defined.
A: Step by step, and naming the properties: $$\begin{array}{llr} B(U+V)B^{-1} &= B\left[(U+V)B^{-1}]\right] & \text{associative} \\ &= B\left[UB^{-1} + VB^{-1}\right] & \text{right distributive} \\ &= BVB^{-1} + BUB^{-1} & \text{left distributive}\end{array}$$
In conjunction with the fact (which you can easily prove) that $B(\lambda A)B^{-1} = \lambda BAB^{-1}$, this indeed makes the map $\gamma_B (A) = BAB^{-1}$ a linear transformation over $\operatorname{GL}_n(\Bbb R)$ (invertible $n\times n$ matrices with coeff. in $\Bbb R$). In fact it preserves products too! This action is called conjugation by the matrix $B$.
