# MATB24H3 Lecture Notes - Orthogonal Complement, Dot Product, Proa

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Published on 21 Apr 2013

School

Department

Course

Professor

University of Toronto at Scarborough

Department of Computer & Mathematical Sciences

MAT B24S Fall 2011

MAT B24

Lectures 14.

1 Orthogonality

1.1 Projections

Review: In R3with the dot product as an inner product: consider the ques-

tion of ”decomposing” a vector binto a sum of two terms: one parallel to a

speciﬁed nonzero vector aand the other perpendicular to a. We will call the

component parallel to the vector athe orthogonal projection of bon aand

denote it by projab. Let p=projab, then we want to ﬁnd pand v= such

that vis orthogonal to aand b=p+v

In A23 we have shown that p=projab=b·a

||a||2aLet us remind ourselves

of the proof:

Suppose that b=p+vwhere p=kaand vis orthogonal to a. Then :

b·a= (p+v)·a=p·a+v·a= (ka)·a+ 0 = k(a·a)

Thus

k=b·a

a·a=b·a

||a||2

and

p=b·a

||a||2a

and v=b−p

1

EXAMPLE: Find p, the orthogonal projection of b= [3,1,−7] on a=

[1,0,5] and v, the vector component of borthogonal to a.

solution. If pis the orthogonal projection of bon a, then:

p=projab=b·a

||a||2a=3·1 + 1 ·0 + (−7) ·5

1 + 25 a=−32

26 [1,0,5] = −16

13 ,0,−80

13

and if vis the vector component of borthogonal to a, then:

v=b−p= [3,1,−7] −−16

13 ,0,−80

13 =23

13,1,−171

13

We will expand the above to Rn

In the previous example, we may think of pas a vector in the subspace

W=sp(a) and of vas a vector from a set of all vectors perpendicular to

every vector in W.

DEFINITION: Suppose that Wis a subspace of the vector space Vwith

inner product <, >, then the set

W⊥={v∈V|<v,w>= 0 ∀w∈W}

is called the orthogonal complement of W.

With the above deﬁnition then, our previous example can be viewed as

b=p+vwhere p∈W=sp(a) and v∈W⊥

Suppose that b∈Rnand Wis a subspace of Rnand with dot product

again as the inner product we get W⊥. We will show that we can ”decom-

pose” binto the sum of two unique vectors bWand bW⊥such that bW∈W

and bW⊥. We will call the vector bWthe projection of bon W.

2

## Document Summary

We will call the component parallel to the vector a the orthogonal projection of b on a and denote it by projab. Let p = projab, then we want to nd p and v = such that v is orthogonal to a and b = p + v. In a23 we have shown that p = projab = ||a||2 a let us remind ourselves of the proof: Suppose that b = p + v where p = ka and v is orthogonal to a. A = (p + v) a = p a + v a = (ka) a + 0 = k(a a) Thus and and v = b p k = Example: find p, the orthogonal projection of b = [3, 1, 7] on a = [1, 0, 5] and v, the vector component of b orthogonal to a. solution. If p is the orthogonal projection of b on a, then: p = projab =