Chapter 5 · Introduction

Chapter 5 — Vector Spaces

Arrows, polynomials, matrices and signals all add and scale by the same rules — so one theory of vector spaces, spans, bases and dimension covers them all.

So far a vector has been an arrow in the plane or a column of numbers. But step back and ask what we have actually done with vectors: we have added them and multiplied them by numbers — nothing else. Now notice how many other objects can be added and scaled in exactly the same way. Two polynomials add to a polynomial; a sound recording played twice as loud is a scalar multiple; two photographs can be blended pixel by pixel; two $2\times2$ matrices add to a $2\times2$ matrix. Mixing two tones on a mixing desk is, mathematically, the same act as adding two arrows nose to tail.

This chapter turns that observation into a theory. We list the rules that addition and scaling must obey and call any set that obeys them a vector space. Then everything we prove — about which vectors can be built from others, which are redundant, how many are needed, how to describe a vector by coordinates — holds at once for arrows, polynomials, matrices, signals and functions. Engineers use these ideas to compress images and audio, statisticians to count the "degrees of freedom" in data, and every solver of $A\mathbf{x}=\mathbf{b}$ uses them to describe all the solutions at once.

Vector-space axioms→Subspaces→Span & linear independence→Basis & dimension→Row, column & null spaces; rank→Coordinates & change of basis
🎯 By the end of this chapter you can…
  • recognise vector spaces such as $\mathbb{R}^n$, $P_n$ and $M_{m\times n}$, and use the subspace test to decide whether a subset is a subspace;
  • decide whether a vector lies in a span, and whether a set spans a space, by solving a linear system;
  • test a set for linear independence by row reduction and find a dependence relation when there is one;
  • find a basis and the dimension of a subspace;
  • compute bases for $\operatorname{Row}A$, $\operatorname{Col}A$ and $\operatorname{Nul}A$ and use $\operatorname{rank}A+\operatorname{nullity}A=n$;
  • find the coordinate vector $[\mathbf{x}]_{\mathcal B}$ and change coordinates with $P_{\mathcal C\leftarrow\mathcal B}$.
Teaser — One Recipe, Many Kinds of Vector

The same linear combination $c_1\mathbf{u}+c_2\mathbf{v}$ is formed in three different vector spaces. Press Play to let the weights $c_1$, $c_2$ change, and switch the space: the green result is built from the blue $\mathbf{u}$ and the red $\mathbf{v}$ in exactly the same way each time.

$c_1$—
$c_2$—
$c_1\mathbf{u}+c_2\mathbf{v}$—

Sections in this chapter

5.1

Vector Spaces and Subspaces

The ten axioms, examples from polynomials to functions, the subspace test, null spaces and spans.

$\mathbf{0}\in W,\ \mathbf{u}+\mathbf{v}\in W,\ c\mathbf{u}\in W$
Open section →5.2

Spanning Sets and Linear Independence

Is a vector in a span? Is any vector redundant? Both become linear systems.

$\sum c_i\mathbf{v}_i=\mathbf{0}\Rightarrow\text{all }c_i=0$
Open section →5.3

Basis and Dimension

Sets that span with nothing wasted; every basis of a space has the same size.

$\dim\mathbb{R}^n=n,\ \dim P_n=n+1$
Open section →5.4

Row Space, Column Space, Null Space and Rank

The subspaces attached to a matrix, and the rank–nullity theorem.

$\operatorname{rank}A+\operatorname{nullity}A=n$
Open section →5.5

Coordinates and Change of Basis

Describe a vector by its weights in a basis, and translate between bases.

$[\mathbf{x}]_{\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}[\mathbf{x}]_{\mathcal B}$
Open section →
✅ Before you start
You should be fluent with row reduction to (reduced) row echelon form and with reading off free variables (1.2 Gaussian and Gauss–Jordan Elimination), with matrix–vector products (2.1 Matrix Operations), and with adding and scaling vectors in $\mathbb{R}^n$ (4.1 Vectors in ℝ², ℝ³ and ℝⁿ). Almost every question in this chapter ends in a row reduction.
← 4.3 Cross Product, Lines and Planes5.1 Vector Spaces and Subspaces →