Subspaces

Let’s say we have:

$$A=\left[\begin{array}{ccc}1& 0& 0\\ -1& 1& 0\\ 0& -1& 1\end{array}\right],C=\left[\begin{array}{ccc}1& 0& -1\\ -1& 1& 0\\ 0& -1& 1\end{array}\right]$$

The columns of $C$ lie in the same plane such that they are dependent, and the columns of $A$ are independent, (see Linear Independence and Vector Combinations).

There are many vectors in ${\mathbb{R}}^3$ that don’t lie in the plane; these vectors cannot be written as a linear combination of the columns of $C$ and so correspond to values of $\vec{b}$ for which $C\vec{x}=\vec{b}$ has no solution $\vec{x}$. The linear combinations of the columns of $C$ form a 2-dimensional subspace of ${\mathbb{R}}^3$.

This plane of combinations of $\vec{u},\vec{v},\vec{w}$ (the columns of $C$) can be described as “all vectors $C\vec{x}$”. But we know that vectors $\vec{b}$ for which $C\vec{x}=\vec{b}$ satisfy the condition $b_1+b_2+b_3=0$ (from this example). So the plane of all combinations of $\vec{u}$ and $\vec{v}$ consists of all vectors whose combinations sum to $0$.

On the other hand, if we take $A$, such that we have columns

$$\vec{u}=\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right],\vec{v}=\left[\begin{array}{c}0\\ 1\\ -1\end{array}\right],\vec{w}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$$

we get the entire space ${\mathbb{R}}^3$; the equation $A\vec{x}=\vec{b}$ has a solution for every $\vec{b}$ in ${\mathbb{R}}^3$. We say that $\vec{u},\vec{v},\vec{w}$ form a basis for ${\mathbb{R}}^3$.

Basis, Vector Space, Subspace

A basis for ${\mathbb{R}}^n$ is a collection of $n$ independent vectors in ${\mathbb{R}}^n$. Equivalently, a basis is a collection of $n$ vectors whose combinations cover the whole space. Or, a collection of vectors forms a basis whenever a matrix has those vectors as its columns is invertible.

• A vector space is a collection of vectors that is closed under linear combinations.
• A subspace is a vector space inside another vector space; a plane through the origin in ${\mathbb{R}}^3$ is an example of a subspace.
  • A subspace could be equal to the space it’s contained in
  • The small space contains only the zero vector $\vec{0}$

The subspaces of ${\mathbb{R}}^3$ are:

• The origin
• A line through the origin
• A plane through the origin
• All of ${\mathbb{R}}^3$