Subspace

For $V\in \Sigma$, we call $V$ a linear subspace of a vector space $\Sigma$ (the space is usually ${\mathbb{R}}^n$) if it satisfies all of the following properties:

1. $V$ contains the neutral element $0$ of $\Sigma$. For ${\mathbb{R}}^n$, this is $\vec{0}$.
2. For any $\overrightarrow{u_1},\overrightarrow{u_2}\in V$, their sum $\overrightarrow{u_1}+\overrightarrow{u_2}$ must also be in $V$, i.e., $V$ is closed under vector addition.
3. For any vector $\vec{u}\in V$ and any scalar $k\in \mathbb{R}$, the vector $k\vec{u}$ must also be in $V$, i.e., $V$ is closed under scalar multiplication.

The fundamental subspaces of linear transformations are the image and kernel.

We define the orthogonal complement as the set of vectors that is orthogonal to the vectors in $V$.

$$V^{\perp }=\{\vec{v}\in {\mathbb{R}}^n:\vec{v}\cdot \vec{w}=0\forall \vec{w}\in V\}$$

Theorems

• For a subspace $V\in {\mathbb{R}}^n$, $\text{dim}(V)+\text{dim}(V^{\perp })=n$.
• Every $\vec{v}\in V$ can be uniquely expressed as $\vec{v}={\vec{v}}^{\Vert }+{\vec{v}}^{\perp }$ where ${\vec{v}}^{\Vert }\in V$ and ${\vec{v}}^{\perp }\in V^{\perp }$.