Orthogonal Sets and Projections in Vector Spaces, Study notes of Mathematics

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Let

For which value of

k^

is

the vector

u

orthogonal to the vector

v

k=

k=-

k=-5/

No such

k

exists.

6.2 Orthogonal Sets A set of vectors

in^

is called an

orthogonal set

if^

whenever

Theorem.

Suppose

is an

orthogonal set of nonzero vectors in

and

Then

is a linearly

independent set and is therefore a basis for Example.

is an

orthogonal basis for

An Orthogonal ProjectionProblem

: For a fixed nonzero vector

in

write

in^

as:

where

is a scalar,is a vector orthogonal to

5

An Orthogonal Projection For a given nonzero vector

in^

we can

decompose

in^

as:

where

(orthogonal projection of

onto

)

(component of

orthogonal to

)

Orthonormal Sets A set of vectors

in^

is called an

orthonormal set

if it is an orthogonal set and for all

If^

then

is an

orthonormal basis

for

Example.

is an

orthogonal basis for

R

3. An orthonormal basis is

is called a

unit vector

if

Suppose

where

is an orthonormal set.Then It can also be shown that

So

Such a matrix is called an

orthogonal matrix

Theorem

. An

mxn

matrix

U

has orthonormal columns

if and only if

U

T U=I

Theorem.

Let

U

be an

mxn

matrix with orthonormal

columns, and let

x

and

y

be in

R

n. Then

13

Theorem. (The Orthogonal Decomposition Theorem

Let

W

be a subspace of

R

n^.^

Then each

y

in

R

n^ can be

uniquely represented in the formwhere

is in

W

and

is in

In fact, if

is any orthogonal basis for

W,

then and

14

Theorem. (The Orthogonal Decomposition Theorem

Let

W

be a subspace of

R

n^.^

Then each

y

in

R

n^ can be

uniquely represented in the formwhere

is in

W

and

is in

In fact, if

is any orthogonal basis for

W,

then andThe vector

is called the

orthogonal projection of

y onto W.

Geometric Picture

Let

Find the

orthogonal projection of

y onto

(a)

(b)

(c)

Example.

Find the closest point to

in

where Solution.