Complementary Subspaces

Let $$V$$ and $$W$$ be subspaces of $$U$$, i.e., $$V,W \subseteq U$$. $$V$$ and $$W$$ are called complementary subspaces if V$$ and $$w \in W$$, and
 * 1) Any $$u \in U$$ can be expressed as $$u=v+w$$ for some $$v \in
 * 1) The intersection of $$V \text{ and } W$$ contains only 0 (which is the requirement to be a subspace), i.e. $$V \cap W = \{0\}$$

Alternative Notation
An alternative and arguably clearer notation for complementary spaces is as follows. Let $$\mathcal{X}$$ and $$\mathcal{Y}$$ be subspaces of $$V$$. If $$\mathcal X$$ and $$\mathcal Y$$ are complementary subspaces, then

$$\mathcal V = \mathcal X \oplus \mathcal Y$$, which means that (as noted above)


 * 1) $$\mathcal X + \mathcal Y = \mathcal V$$ (i.e., the union of the two subspaces is $$\mathcal V$$); and
 * 2) $$X \cap Y = \mathbf 0$$

A quintessential example to complementary subspace comes from the fundamental theorem of linear algebra:

$$\mathcal R(A)$$ and $$\mathcal N(A^T)$$ are complementary subspaces for any matrix $$A \in \mathbb{R}^{n\times k}$$, which can be written as:
 * $$\mathcal R(A) + \mathcal N (A^T) = \mathbb{R}^{n}$$ and $$\mathcal{R}(A)\perp \mathcal N (A^T)$$
 * or $$\mathcal R (A)^\perp = \mathcal N (A^T)$$

Clearly, the same works for the $$A^T$$ as well:
 * $$\mathcal R(A^T) + \mathcal N (A) = \mathbb{R}^{n}$$ and $$\mathcal{R}(A^T)\perp \mathcal N (A)$$
 * or $$\mathcal R (A^T)^\perp = \mathcal N (A)$$