Inner Product spaces Orthogonal sets R2, R3, R4, C4, P(R)

(b) Inner product spaces Orthogonal sets R², R³, R⁴, C⁴, P(R)

Problem 1.

Apply the Gram-Schmidt process to the given subset S = {(1, 0, 1), (0, 1, 1), (1, 3, 3)} and x = (1, 1, 2) of the inner product space V = R³

(a) to obtain an orthogonal basis for span (S)

(b) Normalize the vectors in this basis to obtain an orthonormal basis β for span (S)

(c) Compute the Fourier coefficients of the given vector relative to β.

Solution :

(a) Apply Gram-Schmidt process

(b) The vectors v₁, v₂ and v₃ can be normalized to obtain the orthonormal basis {u₁, u₂, u₃},

where,

Hence, the required orthonormal basis of span(S) is :

(c) Now, find the Fourier coefficients for, x = (1, 1, 2)

Hence, the required Fourier coefficients are :

Problem 2.

Apply the Gram-Schmidt process to the given subset S = {(1, -2, -1, 3), (3, 6, 3, −1), (1, 4, 2, 8)} and x = (-1, 2, 1, 1) of the inner product space V = R⁴ (a) to obtain an orthogonal basis for span (S) (b) Normalize the vectors in this basis to obtain an orthonormal basis β for span (S) (c) Compute the Fourier coefficients of the given vector relative to β.

Solution :

(a) Apply Grass-Schmidt process

(b) The vectors v₁, v₂ and v₃ can be normalized to obtain the orthonormal basis {u₁, u₂, u₃}, where

Thus, the required orthonormal basis of span(S) is,

(c) Next compute the Fourier coefficients for,

Hence, the required of Fourier coefficients are,

Problem 3.

Apply the Gram-Schmidt process to the given subset

of the inner product space V = C⁴

(a) to obtain an orthogonal basis for span (S)

(b) Normalize the vectors in this basis to obtain an orthonormal basis β for span (S)

(c) Compute the Fourier coefficients of the given vector relative to β.

Solution :

(a) Apply Grass-Schmidt process

Similarly,

(b) The vectors v₁ and v₃ can be normalized to obtain the orthonormal basis {u₁, u₂, u₃}, where,

Hence, the required orthonormal basis of span(S) is,

(c) Next, compute the Fourier coefficient for,

Thus, the required Fourier coefficients are,

Problem 4.

In R², let

Find the Fourier coefficients of (3, 4) relative to β

Solution :

Let the set 

The lengths of the vectors in β are,

and

It is also given that the vectors are linearly independent.

Hence, β is an orthonormal basis of R².

If β is an orthnormal subset of an inner product space V and let x Є V, then define the Fourier coefficients of x relatives to β to the scalars (x,y), where y Є β.

.'. Fourier coefficients of (3, 4) are,

.'. The required Fourier coefficients of (3, 4) are and 7/√2 and -1/√2

Problem 5.

Apply the Gram-Schmidt process to the given subset S the inner product space V to obtain

(a) Orthogonal basis for span(S).

(b) Normalize the vectors in this basis to obtain an orthonormal basis β for span(S).

(c) Compute the Fourier co-efficiensis of the given vector relative to β.

Solution:

Consider the vector space V = P₂(R) with inner product defined as,

Let the vectors,

(a) Now, apply Grass-Schmidt process

(b) The vectors v₁, v₂ and v₃ can be normalized to obtain the orthonormal basis {u₁, u₂, u₃}, where

Thus, the required orthonormal basis of span(S) is

(c) Now, compute the Fourier coefficients for,

h(x) = 1 + x

So the coefficient,

Again,

And,

Hence, the required Fourier coefficients are

Problem 6.

Apply the Gram-Schmidt process to the given subset

 of the inner product space V = R³

(a) to obtain an orthogonal basis for span (S)

(b) Normalize the vectors in this basis to obtain an orthonormal basis β for span (S)

(c) Compute the Fourier coefficients of the given vector relative to β.

Solution :

Let the inner product space,

(a) Now, apply Grass-Schmidt process

(b) The vectors v₁ and v₂ can be normalized to obtain the orthonormal basis {u₁, u₂} where,

(c) Next compute the Fourier coefficients for,

Thus, the required Fourier coefficients are,