Question: Let \(H\) and \(K\) be subspaces of \({{\mathbb{R}}^{4}}\) given by

\[H=\left\{ \left[ \begin{matrix} a \\ b \\ c \\ d \\ \end{matrix} \right]\in {{\mathbb{R}}^{4}}:\,\,b=0\text{ and }d=0 \right\}\] \[K=\left\{ \left[ \begin{matrix} a \\ b \\ c \\ d \\ \end{matrix} \right]\in {{\mathbb{R}}^{4}}:\,\,a=0\text{ and }d=0 \right\}\]

Find the dimensions of H and of K. Find a basis for \(H\cap K\).