Inner Product space-Matrices

(a) Inner product space - Matrices

Problem 1.

Apply the Gram-Schmidt process to the given subset S of 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 :

(a) Now apply Gram-Schmidt process

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

The vector u₂ is given by,

Thus, the required orthonormal basis of span(S) is β = {u₁, u₂, u₃}

[Use your calculator for matrix multiplication]

(c) Next compute the Fourier coefficients for,

The coefficient c₁ is given by

The coefficient c₂ is given by

and

The coefficient c₃ is given by

Problem 2.

Apply the Gram-Schmidt process to the given subset S of 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:

(a) Now apply Gram-Schmidt process

(b) The vectors v₁, v₂ and v₃ can be normalized to obtain the orthnormal basis {u₁, u₂, u₃}., where u₁ is given by

The vector u₂

And the coefficient is

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

β = {u₁, u₂, u₃}

(c) Next compute the Fourier coefficients for,

The coefficient c₁ is,

Again coefficient c₂ is,

And,

The coefficient c₃ is,

Thus the required Fourier coefficients are

Problem 3.

Let A be an n x n matrix with complex entries. Prove that AA* = I if and only if the rows of A form an orthonormal basis for C_n.

Solution :

Let A be an n X n matrix with complex entries.

To prove that AA* = I if and only if the rows of A form an orthonormal basis for Cn.

We know that,

By definition of an orthonormal basis.

Suppose that AA* = I.

Hence, all the diagonal entries in the matrix AA* are 1 and all non-diagonal entries are 0.

Therefore, the entries in AA* can be written as,

Thus, the rows of A form an orthonormal basis for Cⁿ

Suppose the rows of A form an orthonormal basis for Cⁿ

Then,

Observe that the entries for the matrix AA* are either 0 or 1.

From the above definition of <v_i, v_j>, observe that all the diagonal entries in the matrix AA* are 1 and all non-diagonal entries are 0.

Thus, AA* = I

Problem 4.

Prove that for any matrix 

Solution :

To prove that for any matrix 

Suppose 

This gives that x is orthogonal to A*y for all y Є F^{m n}

The above relation holds for all y.

Therefore, Ax = 0