The Gram Schmidt Orthogonalization: Theorems Part 1 - Example Solved Problems

THE GRAM SCHMIDT ORTHOGONALIZATION

THEOREMS PART 1

WORKED EXAMPLE PROBLEMS

Example 1

 Let {(1, 1, 0), (1 − 1, 1), (− 1, 1, 2)} be an orthogonal set then orthonormal set is { 1/√2(1, 1, 0), 1/√3 (1, − 1, 1), 1/√6(− 1, 1, 2) } both are basis of R³. Let x=(2, 1, 3) ∈ R³. Express x as a linear combination of orthogonal set S & orthonormal set.

Solution:

By T Theorem 1. We have

⇒ x as a linear combination at the basis vectors are

 a₁ = 1/√2 (2+1) = 3/√2

a₂ = 1/√3 (2‒1+1) = 4/√3

a₃ = 1/√6 (‒2+1+6) = 5/√6

 (2, 1, 3) = [(2+1)/2]  (1,1,0) + 4/3 (1,−1,1) + 5/6 (−1,1,2)

which is a linear combination of orthogonal set.

 x = ^kΣ_i=1 < x, v_i> v_i

= <x, v₁> v₁ + <x, v₂>v₂ + <x, v₃>v₃

 (2, 1, 3) = 3/2 (1, 1, 0) + 4/3 (1, − 1, 1) + 5/6 (− 1, 1, 2)

Example 2

Let R² have the weighted Euclidean inner product defined as <u, v> =2u₁v₁ + 3u₂v₂ and let u = (1, 1), v = (3,2), w = (0,− 1). Compute the value of (u+v, 3w).

Solution:

Given that

< u, v> = 2u₁v₁ + 3u₂v₂

 u+v = (1, 1)+(3,2) = (4, 3) and 3w = 3(0,−1) = (0,−3)

 <u+v, 3w> = 2(4)(0) + 3(3)(−3) = 0+(−27) = −27

Example 3

Let P₂ have the inner product <p,q> = _-1ʃ¹  p(x) q(x) dx. Find the angle between p and q, where p=x and q=x² with respect to the inner product on P₂.

Solution:

Let <p,q> = _-1ʃ¹  p(x) q(x) dx where p =x and q=x².

Let p =x and q = x².

Example 4

Consider V=R³. Let y₁ = (1, 1, 0), y₂ = (2, 0, 1), y₃ = (2, 2, 1). Check {y₁y₂,y₃} is linearly independent, using Gram schmidt process. Compute orthogonal vectors from {y₁, y₂,y₃}.

Solution:

  |Δ| =  = ‒2 ≠ 0

{y₁, y₂, y₃} is linearly independent.

Let {x₁, x₂, x₃} be the orthogonal set.

 <y₂, x₁> = < (2, 0, 1), (1, 1, 0) > = 2+0+0=2 and

||x₁|| = √[1²+1²+0²] = √2

 <y₃, x₁> = < (2, 2, 1), (1, 1, 0) > = 2 +2 = 4

 <y₃, x₂> = < (2, 2, 1), (1, 1, 1) > = 2−2+1=1

||x₂|| = √[1² + (−1)² + 1 ]

= √3

 (x₁, x₂, x₃) is an orthogonal set.

Example 5

Let y₁ = (1, 1, 1); y₂ = (−1, 0, −1); y₃=(−1, 2, 3). Check {y₁ y₂ y₃} is linearly independent using Gram−Schmidt process. Compute orthogonal vectors from {y₁, y₂, y₃}

Solution:

= 4 ≠ 0

Given {y₁, y₂, y₃} is linearly independent. Let { x₁, x₂, x₃ } be orthogonal set.

Example 6

Let V=R⁴, Let w₁ = (1, 0, 1, 0), w₂ = (1, 1, 1, 1), w₃ = (0, 1, 2, 1). Use Gram−Schmidt process to compute the orthogonal vectors & normalize these vectors.

Solution:

Let { x₁, x₂, x₃} be the orthogonal set.

 <w₃, x₁> = < (0, 1, 2, 1), (1, 0, 1, 0) > =0+0+2+0=2

 <w₃, x₂> = < (0, 1, 2, 1), (0, 1, 0, 1) > =0+1+0+1=2

||x₂|| = √[0² + 1² + 0² + 1²] = √2

  {x₁, x₂, x₃} is an orthogonal set.

 || x₁|| = √2; ||x₂|| = √2; ||x₃|| = √2

Orthonormal set is {y₁, y₂, y₃} where

Example 7

Let V=P (R) with the inner product < f(x),g(x) > = _-1ʃ¹ f(t)g(t) dt. Consider the sub space P₂(R) with standard ordered basis β. Use the Gram−Schmidt process to replace β by an orthogonal basis {v₁, v₂, v₃ } for P₂(R) & use those orthogonal basis to obtain an orthonormal basis for P₂(R).

Solution:

{y₁, y₂, y₃} is an orthonormal basis for P₂(R).

Example 8

Let V=M_2×2(R). Let S =  Use Gram−Schmidt process to compute the orthogonal vectors and normalize these vectors. Also compute the Fourier coefficients related to basis.

Solution:

Now the vectors x₁, x₂, x₃ can be normalized to obtain the orthonormal basis {y₁, y₂, y₃} where

Therefore, the required orthonormal basis are B = { y₁, y₂, y₃ }

To find Fourier coefficients:

The coefficient of c₂ is given by c₂ = <A, y₂>

The coefficient of c₃ is given by c₃ =(A, y₃)

Example 9

Let R³ have the Euclidean inner product. Use Gram−Schmidt process to transform the basis {u₁, u₂, u₃} into an orthonormal basis, where u₁ = (1, 1, 1), u₂ = (0, 1, 1) and u₃ = (0, 0, 1).

Solution:

  β = {y₁, y₂, y₃} is the orthonormal basis.

Example 10

Let the vector space P₂ have the inner product <p,q> = ₀ʃ¹ p(x) q(x) dx. Apply the Gram−Schmidt process to transform the basis S={u₁, u₂, u₃}={1,x,x²} and h(x)=1+x into an orthonormal basis and compute the Fourier coefficients.

Solution:

Let S = { u₁, u₂, u₃ } = { 1, x, x² }

The vectors v₁, v₂, v₃ can be normalized to obtain the orthonormal basis { u₁, u₂, u₃ }.

To find the Fourier coefficients

It is given that h(x) = 1+x

Example 11

Let V=R³. Define S={(1, i, 0), (1−i, 2, 4i)} and x= {3+i, 4i, −4}. Apply Gram−Schmidt process to obtain orthogonal basis and normalize the vectors to get orthonormal basis.

Solution:

Let V=R³, Given that S= {(1, i, 0), (1‒i, 2, 4i) }.

Let w₁ = (1, i, 0) and w₂ = (1−i, 2, 4i)

 S={w₁, w₂},

Let v₁ =w₁ = (1, i, 0)

 v₂ = w₂ ‒ { <w₂,v₁>/||v₁||²}v₁

 < w₂, v₁> = < (1 − i, 2, 4i), (1, i, 0) > = (1 − i) (1) + 2(− i) + 4i (0)

= (1−i−2) = (1−3i)

||v₁||² = < v₁, v₁> = < (1, i, 0), (1, i, 0) >

= (1)(1) + (i)(− i) + (0)(0)

 = 1−i²

= 1+1=2

 v₂ = w₂ ‒ { <w₂,v₁>/||v₁||²}v₁

= (1 − i, 2, 4i) ‒  (1-3i)/2 (1,i, 0)

 = (1 − i, 2, 4i) ‒ 1/2(1-3i, i+3, 0)

v₂ = ( 1/2(1+i), 1/2(1‒i), 4i )

= ½ [ 1+i, (1‒i), 8i ]

||v₂||² = <v₂,v₂> = < 1/2(1+i, 1‒i, 8i), ½(1+i, 1‒i, 8i) >

= [ 1/2(1+i) . 1/2(1‒i) + 1/2(1‒i)  . 1/2(1+i) 1/2(8i) . 1/2(‒8i)  ]

= [ 1/4[1-i²] + 1/4(1-i²) + 1/4.64(-i²) ]

= [ 1/2 + ½ + 16 ]

= 17

The vectors v₁ and v₂ can be normalized to obtain the orthonormal basis {y₁, y₂ }.

  y₁ = ( 1/||v₁|| ) v₁ = 1/√2 (1,i,0)

  y₂ = ( 1/||v₂|| ) v₂ = 1/2√17 (1+i,1‒i,8i)

Example 12

Let V=R². Define T (a, b) = (2a + b, a−3b) and x = (3, 5). Evaluate T* for V=R².

Solution:

 Let α= {(1, 0), (0, 1)} be the standard basis for V=R².

It is given that T(a, b) = (2a + b, a − 3b)

 T(1, 0) = (2 (1) +0, 1−3 (0)) = (2, 1)

 T(0, 1) = (2 (0)+1, 0−3 (1)) = (1,−3)

Therefore T = 

Since V=R² is real [T]_α = [T]*_α

The adjoint operator T* at x=(3, 5) is defined as follows.

[T*(x)]_α = [T*]_α [x]_α = [T]^*_α [x]_α = [T]_α [x]_α

 T^* at x = (3, 5) is T*(x) = (11, ‒12)