Deck 10: Inner Product Spaces

Question 1

Evaluate the given inner product on
for

.

Answer 1

48

Question 2

Evaluate the given inner product on
for

.

.

Answer 2

30

Question 3

Evaluate the given inner product on
for

.

Answer 3

1

Question 4

Evaluate the given inner product on
for

.

Question 5

Evaluate

, where the inner product on
is
for

.

Question 6

Evaluate

, where the inner product on
is

.

Question 7

Determine

, where
and

.

Question 8

Determine

, where
and

.

Question 9

Determine

, where
and

.

Question 10

Determine all values of a such that in
the vectors
and
are orthogonal with respect to the inner product

.

Question 11

If
is an inner product on a vector space V, and v is any vector in V, then
defined by
is a linear transformation.

Question 12

If
is an inner product on a vector space V, and
is an invertible linear transformation, then
is also an inner product on V.

Question 13

If u, v are vectors in an inner product space V, then u is orthogonal to v if and only if

.

Question 14

The following is an inner product on

:

Question 15

If
and
are both inner products on a vector space V, then
is also an inner product on V.

Question 16

Let

,

, and let the inner product on
be given by

.
Use the fact that
is an orthogonal basis for S to find

.

Question 17

Let

,

, and let the inner product on
be given by

.
Use the fact that
is an orthonormal basis for S to find

.

Question 18

Find

, where

, where

and the inner product on
is

.

Question 19

Use the Gram-Schmidt process to convert the set
to an orthonormal basis with respect to the inner product

.

Question 20

Use the Gram-Schmidt process to convert the set
to an orthonormal basis with respect to the inner product

.

Question 21

Use the Gram-Schmidt process to convert the set
to an orthonormal basis with respect to the inner product

.

Question 22

Use the Gram-Schmidt process to convert the set
to an orthonormal basis with respect to the inner product

.

Question 23

Let
be a subspace of

. Let

. Find
with respect to the inner product

.

Question 24

Determine the values of a (if any) that will make the given set of vectors orthogonal in
with respect to the inner product

.

Question 25

Determine the values of a (if any) that will make the given set of vectors orthogonal in
with respect to the inner product

.

Question 26

If
is an orthonormal set in an inner product space V, then
is linearly independent.

Question 27

If
is a linear dependent set of nonzero vectors in a vector space V, and
is any inner product on V, then there exists
such that

.

Question 28

If
and
are nonzero subspaces of an inner product space V, and v is any vector in V, then

.

Question 29

If V is a finite-dimensional inner product space, S is a nonzero subspace of V and
is nonzero, then for every
in V,

.

Question 30

If
is an orthonormal set in an inner product space V, and
is a
orthogonal matrix, then
is also an orthonormal set in V.

Question 31

Find the weighted least squares line for the data set

, with the inner three points weighted three times as much as the outer two points.

Question 32

Find the weighted least squares line for the data set

, with the inner three points weighted four times as much as the outer two points.

Question 33

Find the weighted least squares line for the data set

, with the inner point weighted t times as much as the outer two points.

Question 34

Find the Fourier approximation
for

.

Question 35

Find the Fourier approximation
for

.

Question 36

Find the Fourier approximation
for the odd function

.

Question 37

Find the discrete Fourier approximation
for
based on the table information.

Question 38

Find the discrete Fourier approximation
for
based on the table information.

Question 39

Find the Fourier coefficients for the function
without computing any integrals.

Question 40

Find the Fourier coefficients for the function
without computing any integrals.

Question 41

In a weighted least squares regression, the average of the weights must equal 1.

Question 42

If f is a positive even function on

, then for every k, and in the Fourier approximation of f.

Question 43

If function
on

, then the discrete Fourier coefficient

.

Question 44

If f is an odd function on

, then in the Fourier approximation of f, we have
for every k.

Question 45

If
is the nth-order Fourier approximation to f, and
is the nth-order Fourier approximation to g, and
are scalars, then the nth-order Fourier approximation to
is

.