Matrix equality involving transpose

Question

$$\begin{bmatrix} (V_1 \bullet V_1) & (V_2 \bullet V_1)\\ (V_1 \bullet V_2) &(V_2 \bullet V_2) \\ \end{bmatrix} = \begin{bmatrix} V_1\\ V_2\\ \end{bmatrix} \cdot \begin{bmatrix} V_1^T &V_2^T\end{bmatrix} $$

I need to show that this equality is true. So far I tried taking the determinant of each side but I don't know how to relate the two to show the equality holds. Any help would be appreciated.

Answer 1

My answer is based on the assumption that we are dealing with row vectors. Then the right side is simply:

$\begin{pmatrix} V_1V_1^T & V_1V_2^T \\ V_2V_1^T & V_2V_2^T \end {pmatrix} $

which is precisely equal to the left side given the definition of a dot product in Euclidean space.