Chapter 8 — Subspaces of $\mathbb R^n$
A subspace is a set of vectors that behaves like a vector space inside $\mathbb R^n$: it contains zero and is closed under addition and scalar multiplication.
subspace testcolumn spacenullspacerank-nullity
8.1
Subspace test
A subset $W\subseteq\mathbb R^n$ is a subspace if: $0\in W$, $u+v\in W$ whenever $u,v\in W$, and $cu\in W$ whenever $u\in W$ and $c\in\mathbb R$.
Interactive: classify common sets
8.2
Examples and non-examples
Always subspaces
- $\{0\}$
- $\mathbb R^n$
- spans of vectors
- solution sets of $Ax=0$
- column spaces, nullspaces, row spaces
Usually not subspaces
- lines not through the origin
- planes not through the origin
- solution sets of $Ax=b$ with $b\ne0$
- curved sets such as circles or spheres
8.3
Column space and nullspace calculator
The column space is the span of the columns. The nullspace is the solution set of $Ax=0$.
Interactive: compute rank, pivot columns, and nullspace
8.4
Rank-nullity theorem
$$\operatorname{rank}(A)+\operatorname{nullity}(A)=n,$$
where $n$ is the number of columns of $A$.
Interactive: rank-nullity from dimensions
8.5
Geometry of subspaces in $\mathbb R^2$
A line through the origin is closed under addition and scalar multiplication. A shifted line is not: it fails the zero-vector test.
Interactive: line $y=mx+c$
8.6
Chapter summary
- A subspace contains zero and is closed under addition and scalar multiplication.
- Every span is a subspace.
- $\operatorname{Null}(A)$ is the solution set of $Ax=0$.
- $\operatorname{Col}(A)$ is the span of the columns of $A$.
- $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$.
8.7
Quick checks
Why is $y=2x+1$ not a subspace of $\mathbb R^2$?
It does not contain the zero vector.
Why is every span a subspace?
A span contains all linear combinations of its generators, so it is closed under addition and scalar multiplication and contains zero.
What is the nullity of a $4\times 7$ matrix with rank $3$?
$7-3=4$.