3.2 Subspaces · Linear Algebra Refresher

A subspace is a vector space living inside another one, and checking for one is a two-line job instead of eight. Two subspaces matter most: the null space

N (A)

and the span of a set of vectors — and by the end of §3.6 you'll see they're the same kind of object.

Definition · Subspace

A nonempty subset

S

of a vector space

V

is a subspace if it is closed under both operations:

α x \in S

for every scalar

α

and

x \in S

; and

x + y \in S

for all

x, y \in S

. The remaining axioms are inherited from

V

for free.

→Shortcut

Taking

α = 0

in C1 shows

0 \in S

automatically. So the fastest disproof is: does the set contain the zero vector? If not, done in one line.

Always subspaces

{0}

(the zero subspace) and

V

itself — the two trivial subspaces. Also

N (A)

, any span, lines/planes through the origin.

Never subspaces

Anything missing

0

: lines not through the origin,

{x : A x = b}

with

b \neq = 0

, sets defined by

x_{i} \geq 0

x_{1} x_{2} = 0

The null space

Definition · Null space

For an

m \times n

matrix

A

N (A) = {x \in R^{n} : A x = 0}

. It is a subspace of

R^{n}

: if

A x = A y = 0

then

A (α x + y) = α 0 + 0 = 0

Worked example

Finding a basis for $N (A)$

0/5 steps

Determine

N (A)

and write it as a span.

A = (1212 - 1 - 3 21)

Span

Definition · Span

Span (v_{1}, \dots, v_{n})

is the set of all linear combinations

α_{1} v_{1} + \dots + α_{n} v_{n}

. It is always a subspace (Thm 3.2.1). If

Span (v_{1}, \dots, v_{n}) = V

we call

{v_{i}}

a spanning set for

V

Deciding whether $b \in Span (v_{1}, \dots, v_{n})$ is literally the question of whether $V c = b$ is consistent, where $V$ has the $v_{i}$ as columns. Span questions are consistency questions in disguise.

Theorem 3.2.2 · Structure of the solution set

A x = b

is consistent with a particular solution

x_{0}

, then the full solution set is

{x_{0} + z : z \in N (A)}

. The solution is unique iff

N (A) = {0}

This is why the answer to any consistent system looks like 'particular + parameters': the parameters trace out $N (A)$ .

Check your work

Row reduction / rank / null space

Exact fraction arithmetic — type entries like 3, -2, or 1/2. One row per line.

Matrix

1	2	-1	1
2	4	-3	0
1	2	1	5

→

1	2	0	3
0	0	1	2
0	0	0	0

rank

nullity

cols (n)

pivot cols

1, 3

Basis for N(A)

-2

-3

-2

Basis for the column space — original columns 1, 3

-1

-3

Enter any matrix to get its null-space basis, rank, and nullity.

Check yourself

Which subset of

R^{2}

is a subspace?

T / F

The solution set of

A x = b

with

b \neq = 0

is a subspace of

R^{n}

T / F

Span (v_{1}, \dots, v_{n})

is always a subspace, even if the

v_{i}

are linearly dependent.

3.2Subspaces

What you must be able to do

The null space

Finding a basis for $N (A)$

Span

Row reduction / rank / null space

Check yourself

3.2Subspaces

What you must be able to do

The null space

Finding a basis for N(A)

Span

Row reduction / rank / null space

Check yourself

Finding a basis for $N (A)$