5.2 Orthogonal Subspaces · Linear Algebra Refresher

The four subspaces attached to

A

aren't four independent objects — they pair up into two orthogonal complements. This is the geometric completion of everything Chapter 3 said about rank.

Definition · Orthogonal subspaces and complements

Subspaces

X, Y

R^{n}

are orthogonal (

X ⊥ Y

) if

x^{T} y = 0

for every

x \in X

and

y \in Y

. The orthogonal complement of

Y

Y^{⊥} = {x \in R^{n} : x^{T} y = 0 for all y \in Y}

!Exam trap

The

x y

-plane and the

y z

-plane in

R^{3}

are not orthogonal subspaces, even though they meet at right angles. They share the whole

y

-axis, and

e_{2}

is not orthogonal to itself. Orthogonal subspaces intersect only at

0

Theorem 5.2.1 · Fundamental Subspaces

For an

m \times n

matrix

A

N (A) = R (A^{T})^{⊥} \subseteq R^{n}, N (A^{T}) = R (A)^{⊥} \subseteq R^{m} .

The first is almost the definition: $A x = 0$ says $x$ is orthogonal to every row of $A$ , and the rows span $R (A^{T})$ .

Subspace	Lives in	Dimension	Complement
Row space $R (A^{T})$	$R^{n}$	$r$	$N (A)$
Null space $N (A)$	$R^{n}$	$n - r$	$R (A^{T})$
Column space $R (A)$	$R^{m}$	$r$	$N (A^{T})$
Left null space $N (A^{T})$	$R^{m}$	$m - r$	$R (A)$

Two orthogonal splittings:

R^{n} = R (A^{T}) \oplus N (A)

on the domain side,

R^{m} = R (A) \oplus N (A^{T})

on the codomain side.

Theorems 5.2.2–5.2.4

S

is a subspace of

R^{n}

dim S + dim S^{⊥} = n

; every

x \in R^{n}

can be written uniquely as

x = y + z

with

y \in S

z \in S^{⊥}

(the direct sum

R^{n} = S \oplus S^{⊥}

); and

(S^{⊥})^{⊥} = S

Worked example

Computing an orthogonal complement

0/3 steps

Let

S = Span ((1, 0, 1)^{T}, (0, 1, 1)^{T}) \subseteq R^{3}

. Find a basis for

S^{⊥}

∞Connection

Restating consistency geometrically:

A x = b

is consistent iff

b \in R (A)

iff

b ⊥ N (A^{T})

. When it isn't consistent, the best you can do is project

b

onto

R (A)

— which is precisely §5.3.

Check yourself

For an

m \times n

matrix

A

, the orthogonal complement of

N (A)

inside

R^{n}

is:

T / F

The

x y

-plane and the

y z

-plane are orthogonal subspaces of

R^{3}

S

is a

4

-dimensional subspace of

R^{9}

. What is

dim S^{⊥}

5.2Orthogonal Subspaces

What you must be able to do

Computing an orthogonal complement

Check yourself