Math 324: Linear AlgebraSection 5.2: Inner Product Spaces

Mckenzie West

Last Updated: April 17, 2020

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Last Time.

– Dot Product

– Length

– Cauchy-Schwarz Inequality

– Triangle Inequality

Today.

– Orthogonal Complement

– Inner Products

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You may have seen the following definition last time.

Definition.

Two vectors ~u and ~v in Rn are said to be orthogonal if ~u ·~v = 0.

Recall.

The angle between ~u and ~v satisfies

cos(θ) =~u ·~v‖~u‖‖~v‖

.

Thus orthogonal vectors have angle θ, with cos(θ) = 0. Or θ = π2 ,

and are thus perpendicular.

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Exercise 1.

Find all vectors (x , y , z) that are orthogonal to (−2, 1, 0).

(a) Compute (x , y , z) ·(−2, 1, 0).

(b) Set the dot product equal to zero and parameterize.

The equation −2x + y = 0 is equivalent to y = 2x , so if xand z are given parameters, the orthogonal complement of(−2, 1, 0) is the set

{(t, 2t, s) : s, t ∈ R}. (1)

(c) Verify that no matter the values of s and t,(t, 2t, s) ·(−2, 1, 0) = 0.

(d) The set in (1) is called the orthogonal complement of(−2, 1, 0). Plot (−2, 1, 0) and the orthogonal complement inSage (https://sagecell.sagemath.org/) using:s,t = var("s,t")

P = parametric_plot3d((t,2*t,s), (t,-2,2),(s,-2,2))

P += arrow((0,0,0), (-2,1,0), width=3, color="red")

P.show(aspect_ratio=1)

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Exercise 1.

Find all vectors (x , y , z) that are orthogonal to (−2, 1, 0).

(a) Compute (x , y , z) ·(−2, 1, 0).

(b) Set the dot product equal to zero and parameterize.The equation −2x + y = 0 is equivalent to y = 2x , so if xand z are given parameters, the orthogonal complement of(−2, 1, 0) is the set

{(t, 2t, s) : s, t ∈ R}. (1)

(c) Verify that no matter the values of s and t,(t, 2t, s) ·(−2, 1, 0) = 0.

(d) The set in (1) is called the orthogonal complement of(−2, 1, 0). Plot (−2, 1, 0) and the orthogonal complement inSage (https://sagecell.sagemath.org/) using:s,t = var("s,t")

P = parametric_plot3d((t,2*t,s), (t,-2,2),(s,-2,2))

P += arrow((0,0,0), (-2,1,0), width=3, color="red")

P.show(aspect_ratio=1)

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Exercise 1.

Find all vectors (x , y , z) that are orthogonal to (−2, 1, 0).

(a) Compute (x , y , z) ·(−2, 1, 0).

(b) Set the dot product equal to zero and parameterize.The equation −2x + y = 0 is equivalent to y = 2x , so if xand z are given parameters, the orthogonal complement of(−2, 1, 0) is the set

{(t, 2t, s) : s, t ∈ R}. (1)

(c) Verify that no matter the values of s and t,(t, 2t, s) ·(−2, 1, 0) = 0.

(d) The set in (1) is called the orthogonal complement of(−2, 1, 0). Plot (−2, 1, 0) and the orthogonal complement inSage (https://sagecell.sagemath.org/) using:s,t = var("s,t")

P = parametric_plot3d((t,2*t,s), (t,-2,2),(s,-2,2))

P += arrow((0,0,0), (-2,1,0), width=3, color="red")

P.show(aspect_ratio=1)

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Recall.

The four main properties of the dot product are:

1. ~u ·~v = ~v ·~u.

2. ~u ·(~v + ~w) = ~u ·~v + ~u · ~w .

3. c(~u ·~v) = (c~u) ·~v .

4. ~v · ~v ≥ 0 and ~v · ~v = 0 if and only if ~v = ~0.

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Definition.

Let ~u, ~v and ~w be vectors in a vector space V , and let c be anyscalar. An inner product on V is a function that associates areal number 〈~u, ~v〉 for each pair of vectors ~u and ~v and satisfiesthe following axioms.

1. 〈~u, ~v〉 = 〈~v , ~u〉2. 〈~u, ~v + ~w〉 = 〈~u, ~v〉+ 〈~u, ~w〉3. c 〈~u, ~v〉 = 〈c~u, ~v〉4. 〈~v , ~v〉 ≥ 0 and 〈~v , ~v〉 = 0 if and only if ~v = ~0.

A vector space with an inner product is called an inner product

space.

Example.

The dot product is an inner product in Rn, so Rn is an innerproduct space.

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Exercise 2.

Let ~u = (u1, u2) and ~v = (v1, v2) be vectors in R2. Verify that

〈~u, ~v〉 = 3u1v1 + 2u2v2

is an inner product on R2.

(a) Is 〈~u, ~v〉 a real number?

(b) Does 〈~u, ~v〉 = 〈~v , ~u〉?(c) Does 〈~u, ~v + ~w〉 = 〈~u, ~v〉+ 〈~u, ~w〉?(d) Does c 〈~u, ~v〉 = 〈c~u, ~v〉?(e) Is it true that 〈~v , ~v〉 ≥ 0 and 〈~v , ~v〉 = 0 if and only if ~v = ~0?

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Exercise 3.

Let ~u = (u1, u2) and ~v = (v1, v2) be vectors in R2. Use a specificexample to show that

〈~u, ~v〉 = u1v1 − u2v2

is NOT an inner product on R2.That is, pick out an explicit (numerical values) example of ~u and ~vfor which one of the four inner product axioms fails.

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Brain Break.

How many people are in your family?(Define family how you wish. To me, this means parents andsiblings. Maybe you want to countaunts/uncles/cousins/grandparents/etc. Let me know what youdecide.)

This is my family in town for my graduate school graduation.

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Exercise 4.

Let A =

[a11 a12a21 a22

]and B =

[b11 b12b21 b22

]be matrices in the vector

space M2,2. Show that the function

〈A,B〉 = a11b11 + a12b12 + a21b21 + a22b22

is an inner product on M2,2.

Note.

For a finite dimensional vectors space, inner products essentiallyalways look like dot products with positive scalars. Infinitedimensional vector spaces are not always so straight forward.

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Exercise 5.

Let f and g be real valued continuous functions in the vectorspace C [0, 1]. I claim that the following is an inner product,

〈f , g〉 =

∫ 1

0f (x)g(x) dx .

Compute 〈3x + 2, ex〉.(Yes, you can—and should—use your calculator or an onlineintegral calculator).

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Definition.

Let ~u and ~v be vectors in an inner product space V . Then similarlyto in the case of the dot product, we can make similar definitionsfor length, distance, angle, orthogonal, and orthogonal projection.

1. The length (or norm) of ~u is ‖~u‖ =√〈~u, ~u〉.

2. The distance between ~u and ~v is d(~u, ~v) = ‖~u − ~v‖.3. The angle between ~u and ~v is given by

cos θ =〈~u, ~v〉‖~u‖‖~v‖

, 0 ≤ θ ≤ π.

4. We say ~u and ~v are orthogonal if 〈~u, ~v〉 = 0.

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Exercise 6.

Consider the inner product on M2,2 from exercise 4:

〈A,B〉 = a11b11 + a12b12 + a21b21 + a22b22.

Let A =

[1 20 −1

]and B =

[0 52 −2

]. Compute

(a) ‖A‖2

(b) d(A,B)2

(c) Cosine of the angle between A and B.

(d) Are A and B orthogonal?

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Note.

The inequalities also persist for inner products. (Note that inproving the Triangle Inequality for example, we didn’t actually usethe definition of the dot product.)

Theorem 5.8.

Let ~u and ~v be vectors in an inner product space V .

1. Cauchy–Schwarz Inequality: | 〈~u, ~v〉 | ≤ ‖~u‖‖~v‖2. Triangle Inequality: ‖~u + ~v‖ ≤ ‖~u‖+ ‖~v‖3. Pythagorean Theorem: ~u and ~v are orthogonal if and only if

‖~u + ~v‖2 = ‖~u‖2 + ‖~v‖2.

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Note.

One of the great things that we can use this for is to prove theserelationships for integrals without actually talking about integrals!

Exercise 7.

Let f and g be real valued continuous functions in the vectorspace C [0, 1]. Use the inner product,

〈f , g〉 =

∫ 1

0f (x)g(x) dx .

Verify the Cauchy-Schwarz Inequality for f = 3x + 2 and g = ex .(Yes, you can—and should—use your calculator).