Quizmaster

Learning objectives

What you will be able to explain

Norm axioms
Inner product calculation
Inner-product properties
Length and distance
Angle from an inner product
Orthonormal bases

Section 01

Norm axioms to Inner-product properties

Norm axioms

Norms abstract length through three axioms.

Guided checkpoint

Match axiom to statement.

Source: Sec. 3.1, pp. 71-72

Inner product calculation

The dot product is

3+0-4=-1

Guided checkpoint

For $x=[1,2,-1]^T$ and $y=[3,0,4]^T$ , compute $x^Ty$ .

Source: Sec. 3.2, pp. 72-75

Inner-product properties

Inner-product norms obey the parallelogram law; not all norms do.

Guided checkpoint

For a real inner product.

Source: Sec. 3.2, pp. 72-75

Section 02

Length and distance to Orthonormal bases

Length and distance

The displacement is

(3,4)

with length 5.

Guided checkpoint

Compute Euclidean distance between $(1,2)$ and $(4,6)$ .

Source: Sec. 3.3, p. 75

Angle from an inner product

Cosine is the normalized inner product, hence zero.

Guided checkpoint

If nonzero vectors have inner product zero, their angle is:

Source: Sec. 3.4, pp. 76-78

Orthonormal bases

Orthonormality gives immediate coordinates and stable geometry.

Guided checkpoint

Let columns of $Q$ form an ONB.

Source: Sec. 3.5, pp. 78-79

Section 03

Orthogonal complements to Projection coefficient

Orthogonal complements

Projection decomposes a vector into subspace and orthogonal residual components.

Guided checkpoint

Match relationship.

Source: Secs. 3.6 and 3.8, pp. 79-91

Inner product of functions

The integral of

x

from 0 to 1 is

1/2

Guided checkpoint

Using $<f,g\geq int_0^1 f(x)g(x)dx$ , compute $<x,1>$ .

Source: Sec. 3.7, p. 80

Projection coefficient

The coefficient enforces an orthogonal residual.

Guided checkpoint

Projection of $x$ onto nonzero $b$ is:

Source: Sec. 3.8, pp. 81-91

Section 04

Rotations to Exercise-style projection

Rotations

Reflections are orthogonal but have determinant -1.

Guided checkpoint

For a proper rotation matrix $R$ .

Source: Sec. 3.9, pp. 91-94

Further-reading bridge

Functional inner products point toward Hilbert-space geometry.

Guided checkpoint

Which broader framework generalizes lengths and angles beyond Euclidean coordinates?

Source: Sec. 3.10, p. 94

Exercise-style projection

Projection is

[3,0]^T

; residual

[0,4]^T

has norm 4.

Guided checkpoint

Project $x=[3,4]^T$ onto the span of $b=[1,0]^T$ . What is the residual norm?

Source: Chapter 3 exercises, pp. 96-97

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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