Let $\displaystyle V = M_{22}$ the set of all 2 x 2 matrices. Let the operation of vector addition in V be the usual matrix addition but let scalar multiplication in V be defined by:

$\displaystyle c \cdot A = cA^{T}$

Show that V is not a vector space by demonstrating with an example that one of the properties in the definition of a vector space fails to hold.

My attempt:

property I believe fails to hold (let r be any real number):

$\displaystyle r (A+B) = r(A) + r(B)$

I'll define both A and B as the 2 x 2 matrices:

$\displaystyle A=\begin{bmatrix}a & b\\c & d \end{bmatrix}$ and $\displaystyle B=\begin{bmatrix}e & f\\g & h \end{bmatrix}$

$\displaystyle r\begin{bmatrix}a & b\\c & d \end{bmatrix}^{T} + r\begin{bmatrix}e & f\\g & h \end{bmatrix}^{T}$ => $\displaystyle r\begin{bmatrix}a & c\\b & d \end{bmatrix} + r\begin{bmatrix}e & g\\f & h \end{bmatrix}$=>$\displaystyle \begin{bmatrix}ra & rb\\rc & rd \end{bmatrix} + \begin{bmatrix}re & rf\\rg & rh \end{bmatrix}$=> $\displaystyle \begin{bmatrix}ra+re & rb+rf\\rc+rg & rd+rh \end{bmatrix} \notin M_{22}$

Hence, V is not a vector space since it does not hold for the distributivity of scalar multiplication with respect to vector addition. Did I go about this proof correctly?