Consider $M_{2X2}$ such that $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \oplus \begin{pmatrix} e & f \\ g & h \end{pmatrix}=$$ $$\begin{pmatrix} ae & bf \\ cg & dh \end{pmatrix}$$ And constant k times matrix:
$K \otimes \begin{pmatrix} a & b \\ c & d \end{pmatrix}=$ $\begin{pmatrix} a^k & b^k \\ c^k & d^k \end{pmatrix}$ Show that it is a vector space.
I tried by myself to prove the 8 properties to validate the vector space. How does it work when direct sum symbol gets involved with vector spaces? Thanks in advance.
The "direct sum" is a red herring. Since $\oplus$ and $\otimes$ work on each position of the $2×2$ matrix separately, we might as well only consider one such position: $$a\oplus b=ab\qquad k\otimes a=a^k$$ It is clear that this "reduced" structure is also a vector space, with identity $1$ and inverse $1/a$, which means that the original structure is also a vector space.
Note that all of the matrix entries must be positive for the given structure to be a vector space, but this has not explicitly been written in the question.