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Section7.2Orthogonal Complements

Objectives
  1. Understand the basic properties of orthogonal complements.
  2. Recipes: shortcuts for computing the orthogonal complements of common subspaces.
  3. Picture: orthogonal complements in R 2 and R 3 .
  4. Vocabulary words: orthogonal complement, row space.

It will be important to compute the set of all vectors that are orthogonal to a given set of vectors. It turns out that a vector is orthogonal to a set of vectors if and only if it is orthogonal to the span of those vectors, which is a subspace, so we restrict ourselves to the case of subspaces.

Definition

Let W be a subspace of R n . Its orthogonal complement is the subspace

W = A v in R n | v · w = 0forall w in W B .

The symbol W is read W perp”.

This is the set of all vectors v in R n that are orthogonal to all of the vectors in W . We will show below that W is indeed a subspace.

Note

We now have two similar-looking pieces of notation:

A T isthetransposeofamatrix A . W istheorthogonalcomplementofasubspace W .

Try not to confuse the two.

Pictures of orthogonal complements

The orthogonal complement of a line W through the origin in R 2 is the perpendicular line W .

W W

The orthogonal complement of a line W in R 3 is the perpendicular plane W .

W W

The orthogonal complement of a plane W in R 3 is the perpendicular line W .

W W

We see in the above pictures that ( W ) = W .

Example

The orthogonal complement of R n is { 0 } , since the zero vector is the only vector that is orthogonal to all of the vectors in R n .

For the same reason, we have { 0 } = R n .

Since any subspace is a span, the following proposition gives a recipe for computing the orthogonal complement of any subspace. However, below we will give several shortcuts for computing the orthogonal complements of other common kinds of subspaces.

By the proposition, computing the orthogonal complement of a span means solving a system of linear equations. For example, if

v 1 = D 172 E v 2 = D 231 E

then Span { v 1 , v 2 } is the solution set of the homogeneous linear system associated to the matrix

S v T 1 v T 2 T = S 172 231 T .

This is the solution set of the system of equations

U x 1 + 7 x 2 + 2 x 3 = 0 2 x 1 + 3 x 2 + x 3 = 0.

In order to find shortcuts for computing orthogonal complements, we need the following basic facts.

See these paragraphs  for pictures of the second property. As for the third: for example, if W is a (2 -dimensional) plane in R 4 , then W is another (2 -dimensional) plane. Explicitly, we have

Span A e 1 , e 2 B = HNLNJFQO xyzw GRP in R 4 CCCCFQO xyzw GRP · FQO 1000 GRP = 0and FQO xyzw GRPFQO 0100 GRP = 0 INMNK = HNLNJFQO 00 zw GRP in R 4 INMNK = Span A e 3 , e 4 } :

the orthogonal complement of the xy -plane is the zw -plane.

Definition

The row space of a matrix A is the span of the rows of A , and is denoted Row ( A ) .

If A is an m × n matrix, then the rows of A are vectors with n entries, so Row ( A ) is a subspace of R n . Equivalently, since the rows of A are the columns of A T , the row space of A is the column space of A T :

Row ( A )= Col ( A T ) .

We showed in the above proposition that if A has rows v T 1 , v T 2 ,..., v Tm , then

Row ( A ) = Span { v 1 , v 2 ,..., v m } = Nul ( A ) .

Taking orthogonal complements of both sides and using the second fact gives

Row ( A )= Nul ( A ) .

Replacing A by A T and remembering that Row ( A )= Col ( A T ) gives

Col ( A ) = Nul ( A T ) andCol ( A )= Nul ( A T ) .

To summarize:

Recipes: Shortcuts for computing orthogonal complements

For any vectors v 1 , v 2 ,..., v m , we have

Span { v 1 , v 2 ,..., v m } = Nul FQQO v T 1 v T 2 ... v Tm GRRP .

For any matrix A , we have

Row ( A ) = Nul ( A ) Nul ( A ) = Row ( A ) Col ( A ) = Nul ( A T ) Nul ( A T ) = Col ( A ) .