Inner product space

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Section 6.4 Inner product space

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Subsection 6.4.1 Abstract Vector Space

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Let $P_{n} (x) = {a_{0} + a_{1} x + \dots + a_{n} x^{n} | a_{i} \in R}$ is a vector space with two operations

$\begin{aligned} (a_{0} + a_{1} x & + \dots + a_{n} x^{n}) + (b_{0} + b_{1} x + \dots + b_{n} x^{n}) \\ = (a_{0} + b_{0}) + (a_{1} + b_{1}) x + \dots + (a_{n} + b_{n}) x^{n} \end{aligned}$

$k (a_{0} + a_{1} x + \dots + a_{n} x^{n}) = (k a_{0}) + (k a_{1}) x + \dots + (k a_{n}) x^{n}$
Define $C [a, b]$ be the set of all real-valued continuous functions defined on the interval $[a, b] .$ $C [a, b]$ is a vector space with operations

$(f + g) (x) = f (x) + g (x) and (c f) (x) = c [f (x)] .$

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Subsection 6.4.2 Inner Product Space

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Let

u, v,

and

w

be vectors in a vector space

V,

and let

c

be any scalar. An inner product on

V

is a function that associates a real number

⟨ u, v ⟩

with each pair of vectors

u

and

v

and satisfies the axioms listed below.

$⟨ u, v ⟩ = ⟨ v, u ⟩$
$⟨ u, v + w ⟩ = ⟨ u, v ⟩ + ⟨ u, w ⟩$
$c ⟨ u, v ⟩ = ⟨ c u, v ⟩$
$⟨ v, v ⟩ \geq 0,$ and $⟨ v, v ⟩ = 0$ if and only if $v = 0 .$

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Definition 6.4.1. Inner Product Space.

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A vector space

V

with an inner product is called an inner product space.

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Lemma 6.4.2.

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In $P_{n} (x),$ the function

$⟨ p_{1} (x), p_{2} (x) ⟩ = a_{0} b_{0} + a_{1} b_{1} + \dots + a_{n} b_{n}$

is an inner product of $P_{n} (x),$ where $p_{1} (x) = a_{0} + a_{1} x + \dots + a_{n} x^{n}, p_{2} (x) = b_{0} + b_{1} x + \dots + b_{n} x^{n} .$
In $C [a, b],$ the function

$⟨ f (x), g (x) ⟩ = \int_{a}^{b} f (x) g (x) d x$

is an inner product of $C [a, b] .$

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Example 6.4.3.

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C [1, 4],

find the angle between

f (x) = x^{2} - x + 1

and

g (x) = 5 x + 6 .


    
        
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f = x^2-x+1
2
g = 5*x+6
3
f_inner_g = integral(f*g,x,1,4)
4
f_normal = sqrt(integral(f*f,x,1,4))
5
# compute the norm of the function g
6
7
# find the angle between f and g.

    
    
    
    
        
            
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1
# convert the angle from radians to degrees.

    
    
    
    
        
            
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Subsection 6.4.3 Gram-Schmidt Process in an Inner Product Space

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Proposition 6.4.4.

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$P_{n} (x)$ with the inner product in Lemma 6.4.2 is an inner product space.
$C [a, b]$ with the inner product in Lemma 6.4.2 is an inner product space.

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Theorem 6.4.5.

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Let

S = {v_{1}, v_{2}, \dots, v_{n}}

be a basis of an inner product space

V .

Let $B^{'} = {w_{1}, w_{2}, \dots, w_{n}},$ where

$\begin{aligned} w_{1} & = v_{1} \\ w_{2} & = v_{2} - \frac{⟨ v_{2}, w_{1} ⟩}{⟨ w_{1}, w_{1} ⟩} w_{1} \\ w_{3} & = v_{3} - \frac{⟨ v_{3}, w_{1} ⟩}{⟨ w_{1}, w_{1} ⟩} w_{1} - \frac{⟨ v_{3}, w_{2} ⟩}{⟨ w_{2}, w_{2} ⟩} w_{2} \\ ⋮ \\ w_{m} & = v_{n} - \frac{⟨ v_{n}, w_{1} ⟩}{⟨ w_{1}, w_{1} ⟩} w_{1} - \frac{⟨ v_{n}, w_{2} ⟩}{⟨ w_{2}, w_{2} ⟩} w_{2} - \dots - \frac{⟨ v_{n}, w_{n - 1} ⟩}{⟨ w_{n - 1}, w_{n - 1} ⟩} w_{n - 1} \end{aligned}$

$B^{'}$ is an orthogonal basis of $W .$
Let $u_{i} = \frac{w_{i}}{‖ w_{i} ‖} .$ Then $B^{''} = {u_{1}, u_{2}, \dots, u_{n}}$ is an orthonormal basis for $W .$ Also, for $k = 1, 2, \dots, m,$

$span {v_{1}, v_{2}, \dots, v_{k}} = span {u_{1}, u_{2}, \dots, u_{k}}$

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Example: In

C [0, 2],

find an orthonormal basis of the space

W = span {1, x, x^{2}} .

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