8. Vector Spaces

Definition

Let $K$ be a field.

A vector space over $K$ is a set $V$ with:

1. vector addition

\(+:V\times V\to V,\) making $(V,+)$ an abelian group;

1. scalar multiplication

\(K\times V\to V, \qquad (\lambda,v)\mapsto \lambda v,\) satisfying compatibility axioms.

The most important scalar axioms are:

\(\lambda(u+v)=\lambda u+\lambda v,\) \((\lambda+\mu)v=\lambda v+\mu v,\) \((\lambda\mu)v=\lambda(\mu v),\) \(1v=v.\)

Central intuition

A vector space is a world of objects that can be:

• added together;
• scaled by elements of a field;
• combined into linear combinations.

The most important expression in a vector space is

\(\lambda_1v_1+\cdots+\lambda_nv_n.\) So:

A vector space is a space of linear combinations.

Why two worlds appear

In a vector space, there are two kinds of objects:

1. vectors $v\in V$;
2. scalars $\lambda\in K$.

Vectors are the things being combined.

Scalars are coefficients controlling the combination.

A useful slogan:

Vectors are the objects; scalars are the controls.

Why vector addition is abelian

Vector addition represents combining independent contributions.

Examples:

• adding displacements;
• adding forces;
• adding functions;
• adding signals;
• adding columns of numbers.

The order of contributions usually does not matter:

\(u+v=v+u.\) That is why $(V,+)$ is an abelian group.

What scalar multiplication adds

An abelian group lets us add and subtract.

A vector space lets us also say:

• take half of a vector;
• multiply a vector by $3$;
• take $2u-5v$;
• express a vector in coordinates;
• solve linear equations with coefficients.

The field provides a rich arithmetic of coefficients.

Why the distributive axioms appear

There are two additions:

1. vector addition $u+v$;
2. scalar addition $\lambda+\mu$.

Scalar multiplication must respect both.

Scaling a vector sum

\(\lambda(u+v)=\lambda u+\lambda v.\) Meaning:

Scaling a sum of vectors equals the sum of the scaled parts.

Splitting a scalar

\((\lambda+\mu)v=\lambda v+\mu v.\) Meaning:

Applying a sum of coefficients to a vector equals adding the separately scaled copies.

These two laws are the vector-space version of distributivity.

Why the associativity of scalar multiplication matters

\((\lambda\mu)v=\lambda(\mu v).\) Meaning:

Scaling first by $\mu$, then by $\lambda$, is the same as scaling once by $\lambda\mu$.

This ensures scalar multiplication really behaves like scaling.

Why $1v=v$

The scalar $1$ is the neutral scale.

Multiplying by $1$ should not change a vector.

Natural examples

Coordinate spaces

\(K^n\) with componentwise addition and scalar multiplication is the standard example.

Functions

Functions $X\to K$ form a vector space under pointwise operations.

Polynomials

The set $K[x]$ of polynomials is a vector space over $K$.

A polynomial is a linear combination of monomials:

\(a_0+a_1x+a_2x^2+\cdots+a_nx^n.\)

Matrices

All $m\times n$ matrices over $K$ form a vector space.

Solutions to homogeneous linear equations

The solution set of a homogeneous linear system is a vector space.

What vector spaces make possible

Vector spaces are the natural setting for:

• linear combinations;
• span;
• linear independence;
• basis;
• dimension;
• coordinates;
• linear maps;
• matrices;
• eigenvalues and eigenvectors;
• systems of linear equations.

Vector spaces versus modules

If scalars come from a general ring rather than a field, the analogous structure is called a module.

Every vector space is a module over a field, but modules over general rings can behave much less cleanly.

The field assumption gives division by nonzero scalars, which makes basis and dimension theory much nicer.

Place in the build-up

A vector space combines:

• an abelian group of vectors;
• a field of scalars;
• a compatible action of scalars on vectors.

It is not a ring, because vectors do not necessarily multiply with each other.

The next step is to add such an internal multiplication. That leads to algebras over fields.