Evanalysis
9.1Estimated reading time: 9 min

9.1 Inner products, norms, and angles

Define the standard inner product and norm on R^m, then connect those formulas to length, angle, and the first structural inequalities.

Course contents

MATH1030: Linear algebra I

Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.

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All note collectionsSeries overview
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Chapter 1Systems of equations1 sections
1.11.1 Equations and solution sets
Chapter 2Matrices and elimination4 sections
2.12.1 Matrix basics2.22.2 Augmented matrices and row operations2.32.3 Gaussian elimination and RREF2.42.4 Solution-set types
Chapter 3Matrix algebra2 sections
3.13.1 Matrix multiplication and identity matrices3.23.2 Transpose and special matrices
Chapter 4Solution structure1 sections
4.14.1 Homogeneous systems and null space
Chapter 5Invertibility1 sections
5.15.1 Invertible matrices
Chapter 6Vector spaces6 sections
6.16.1 Vector spaces6.26.2 Subspaces6.36.3 Linear combinations and span6.46.4 Linear dependence and independence6.56.5 Basis and dimension6.66.6 Column space, row space, and rank
Chapter 7Determinants3 sections
7.17.1 Determinants and cofactor expansion7.27.2 Row operations, products, and invertibility7.37.3 Transpose, column operations, and Cramer's rule
Chapter 8Eigenvalues and diagonalization3 sections
8.18.1 Eigenvalues, eigenvectors, and eigenspaces8.28.2 Diagonalization and similarity8.38.3 Characteristic polynomials and diagonalization tests
Chapter 9Inner products and orthogonality4 sections
9.19.1 Inner products, norms, and anglesCurrent section9.29.2 Orthogonal sets and orthonormal bases9.39.3 Gram-Schmidt orthogonalization9.49.4 Cauchy-Schwarz and triangle inequalities

Up to this point, most of the course has studied linear structure: span, independence, basis, eigenvectors, and diagonalization. Inner products add geometry. They let you talk about length, perpendicularity, and angle inside the same vector spaces.

The key idea is that geometry can be encoded algebraically by one scalar-valued operation.

Why this section matters

When you write

⟨v,w⟩=vTw,\langle v,w\rangle=v^Tw,

you are not introducing a new symbol for decoration. You are introducing the operation that later controls orthogonality, projection, orthonormal bases, and the inequalities that make Euclidean geometry work in Rm\mathbb{R}^m.

Definition

Standard inner product

For vectors v,w∈Rmv,w\in\mathbb{R}^m, the inner product of v and w is

⟨v,w⟩=∑i=1mviwi=vTw.\langle v,w\rangle =\sum_{i=1}^m v_iw_i =v^Tw.

It is also called the dot product.

Worked example

Compute inner products directly

If

v=[12],w=[34],v= \begin{bmatrix} 1\\2 \end{bmatrix}, \qquad w= \begin{bmatrix} 3\\4 \end{bmatrix},

then

⟨v,w⟩=1⋅3+2⋅4=11.\langle v,w\rangle=1\cdot3+2\cdot4=11.

If

u=[123],z=[456],u= \begin{bmatrix} 1\\2\\3 \end{bmatrix}, \qquad z= \begin{bmatrix} 4\\5\\6 \end{bmatrix},

then

⟨u,z⟩=1⋅4+2⋅5+3⋅6=32.\langle u,z\rangle=1\cdot4+2\cdot5+3\cdot6=32.

Basic inner-product properties

Theorem

Linearity, symmetry, and positivity

For vectors u,v,w∈Rmu,v,w\in\mathbb{R}^m and scalars α,β∈R\alpha,\beta\in\mathbb{R}, the standard inner product satisfies:

⟨v+w,u⟩=⟨v,u⟩+⟨w,u⟩;\langle v+w,u\rangle=\langle v,u\rangle+\langle w,u\rangle;
⟨αv,w⟩=α⟨v,w⟩;\langle \alpha v,w\rangle=\alpha\langle v,w\rangle;
⟨v,w⟩=⟨w,v⟩;\langle v,w\rangle=\langle w,v\rangle;
⟨v,v⟩≥0,\langle v,v\rangle\ge0,

and equality holds if and only if v=0v=0.

The first two properties say the inner product is linear. The third says it is symmetric. The fourth says the inner product measures genuine size rather than an arbitrary signed quantity.

Proof

Why positivity is the key structural axiom

Norm and unit vectors

Definition

Norm

For v∈Rmv\in\mathbb{R}^m, the norm of v is

∥v∥=⟨v,v⟩.\|v\|=\sqrt{\langle v,v\rangle}.

So the norm is the length induced by the inner product.

Theorem

Basic properties of the norm

For every v∈Rmv\in\mathbb{R}^m and every scalar α∈R\alpha\in\mathbb{R}:

∥v∥≥0,\|v\|\ge0,

and ∥v∥=0\|v\|=0 if and only if v=0v=0;

∥αv∥=∣α∣ ∥v∥.\|\alpha v\|=|\alpha|\,\|v\|.

Definition

Unit vector

A vector v is a unit vector if

∥v∥=1.\|v\|=1.

If v≠0v\neq0, then

v∥v∥\frac{v}{\|v\|}

is the normalization of v to a unit vector.

Worked example

Compute a norm and normalize

Let

v=[123].v= \begin{bmatrix} 1\\2\\3 \end{bmatrix}.

Then

∥v∥=12+22+32=14.\|v\|=\sqrt{1^2+2^2+3^2}=\sqrt{14}.

So the corresponding unit vector is

114[123].\frac{1}{\sqrt{14}} \begin{bmatrix} 1\\2\\3 \end{bmatrix}.

Inner product and norm determine each other

The inner product determines the norm immediately through ∥v∥2=⟨v,v⟩\|v\|^2=\langle v,v\rangle. The reverse relationship is also important.

Theorem

Conversion formulas

For vectors u,v∈Rmu,v\in\mathbb{R}^m:

∥u±v∥2=∥u∥2+∥v∥2±2⟨u,v⟩;\|u\pm v\|^2=\|u\|^2+\|v\|^2\pm2\langle u,v\rangle;
  1. the polarization identity is
⟨u,v⟩=12(∥u+v∥2−∥u∥2−∥v∥2);\langle u,v\rangle=\frac12\bigl(\|u+v\|^2-\|u\|^2-\|v\|^2\bigr);
  1. the parallelogram identity is
∥u+v∥2+∥u−v∥2=2∥u∥2+2∥v∥2.\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2.

These formulas explain why inner-product geometry is so rigid: once you know how length behaves, the inner product can be recovered from it.

Angles between vectors

In R2\mathbb{R}^2 and R3\mathbb{R}^3, the inner product can be rewritten in the familiar geometric form

⟨x,y⟩=∥x∥ ∥y∥cos⁡θ,\langle x,y\rangle=\|x\|\,\|y\|\cos\theta,

where θ\theta is the angle between the nonzero vectors x and y.

So

θ=arccos⁡ ⁣(⟨x,y⟩∥x∥ ∥y∥).\theta=\arccos\!\left(\frac{\langle x,y\rangle}{\|x\|\,\|y\|}\right).

In particular, x and y are perpendicular exactly when

⟨x,y⟩=0.\langle x,y\rangle=0.

This observation becomes the definition of orthogonality in arbitrary Rm\mathbb{R}^m.

Common mistake

Common mistake

The inner product is a number, not a vector

Students sometimes see vTwv^Tw and think the result is another vector because it comes from multiplying vectors. It is not. The inner product always produces a single scalar. That is why it can be used to measure angle and length.

Quick check

Quick check

What is ⟨(1,2),(3,4)⟩\langle (1,2),(3,4)\rangle?

Multiply matching coordinates and add.

Solution

Answer

Quick check

If ∥v∥=0\|v\|=0, what must v be?

Use the positivity theorem for the norm.

Solution

Answer

Quick check

If v≠0v\neq0, why is v/\|v\| a unit vector?

Apply the scalar rule for norms.

Solution

Answer

Exercises

Quick check

Compute the norm of [34]\begin{bmatrix}3\\4\end{bmatrix}.

Use ∥v∥=⟨v,v⟩\|v\|=\sqrt{\langle v,v\rangle}.

Solution

Guided solution

Quick check

Normalize [2−21]\begin{bmatrix}2\\-2\\1\end{bmatrix} to a unit vector.

First compute its norm.

Solution

Guided solution

Quick check

Suppose nonzero vectors x and y satisfy ⟨x,y⟩=0\langle x,y\rangle=0. What can you say about their angle in R2\mathbb{R}^2 or R3\mathbb{R}^3?

Use the cosine formula.

Solution

Guided solution

Related notes

Continue with 9.2 Orthogonal sets and orthonormal bases, where inner products are used to build useful bases and coordinate formulas.

Later inequalities in 9.4 Cauchy-Schwarz and triangle inequalities depend directly on the definitions introduced here.

This chapter also continues the geometric thread opened in 6.5 Basis and dimension.

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Skills: inner-product, orthogonal

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Prerequisites

6.1

6.1 Vector spaces

Key terms in this unit

More notes in this series

Previous section

8.3 Characteristic polynomials and diagonalization tests

Next section

9.2 Orthogonal sets and orthonormal bases