Subspace

Flat spaces (lines, planes, volumes...) through the origin are subspaces.

A non-empty subset $V\subℝ^n$ is called a subspace if $\forall\vec{u},\vec{v}\in V, \forall k ∈ ℝ:$

1. $\vec{u}+\vec{v}\in V$
2. $k\vec{u}\in V$

Subspace Properties

• Every subspace is a span and every span is a subspace.

  $V\subℝ^n$ is a subspace iff $V=\text{span }S$ for some set $S$

• Trivial Subspace: Subset $\{\vec{0}\}\sub ℝ^n$ is called the trivial subspace.

• A subspace is closed under addition and scalar multiplication.

Basis

A basis for a subspace $V$ is a linearly independent set of vectors $B$ so that $\text{span }B =V$

E.g. One Basis for $\text{span}\{\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1\\0 \end{bmatrix},\begin{bmatrix} 1\\1 \end{bmatrix}\}$ is $\{\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1\\0 \end{bmatrix}\}$

1. Not unique: Every subspace except the trivial subspace has multiple bases.
2. Given a basis for subspace $V$ every $\vec{v}\in V$ can be written as a unique linear combination of vectors in that basis.
3. Any two bases for the same subspace have the same number of elements.

• How to find Basis in a span:

Standard Basis

The standard basis $\mathcal{E}$ for $ℝ^n$ is the set $\{\vec{e}_1,\dots,\vec{e}_n\}$ where

$\vec{e}_1=\begin{bmatrix} 1 \\ 0\\ \vdots \end{bmatrix}, \vec{e}_2= \begin{bmatrix} 0 \\ 1 \\ \vdots \end{bmatrix}$

($\vec{e}_i$ is the vector with a 1 in its $i$th coordinate and zeros elsewhere)