Suppose $V$ and $W$ are subspaces of $\mathbb{R}^n$ where $V\cap W=\{\vec{0}\}$ and $\{\vec{v}_1,\dots,\vec{v}_p\}$ is a basis for $V$ and $\{\vec{w}_1,\dots,\vec{w}_q\}$ is a basis for $W$.

If:

$$Z=Span(\{\vec{v}_1,\dots,\vec{v}_p,\vec{w}_1,\dots,\vec{w}_q\})$$

...then: