Vector Spaces: Basic Concepts

Abstraction of Geometry

Arrows in space can be added and multiplied by numbers. These operations obey certain laws. Abstracting these laws yields the concept of a vector space.

A vector space over a field F is a set V with addition (V×V→V) and scalar multiplication (F×V→V) operations that satisfy 8 axioms (associativity, commutativity of addition, neutral element, inverse, distributivity, etc.).

Examples

ℝⁿ — the standard example: n-dimensional column vectors.

ℝ^(m×n) — matrices of size m×n. Dim = mn.

The space of polynomials of degree ≤ n: Pₙ. Dim = n+1. Basis: {1, x, x², ..., xⁿ}.

C[a,b] — continuous functions on [a,b]: infinite-dimensional space.

The space of solutions to a linear ODE L[y] = 0 of degree n: n-dimensional.

Linear Dependence and Independence

Vectors v₁, ..., vₖ are linearly dependent if there exist not all zero α₁, ..., αₖ such that α₁v₁ + ... + αₖvₖ = 0.

Otherwise, they are linearly independent.

Check: system Av = 0, where A = [v₁ ... vₖ]. Dependent ⟺ nonzero solution ⟺ rank < k.

Basis and Dimension

A basis B = {b₁, ..., bₙ} is a linearly independent set that generates V.

Any two bases of a space have the same number of elements — the dimension dim V.

Every vector v is uniquely decomposed with respect to a basis: v = α₁b₁ + ... + αₙbₙ. The coefficients αᵢ are the coordinates of v in basis B.

Completion to a basis: any linearly independent set can be extended to a basis.

Subspaces

A subspace U ≤ V is a nonempty subset closed under addition and scalar multiplication.

The intersection of subspaces is a subspace. Sum: U + W = {u+w: u∈U, w∈W}.

Dimension formula: dim(U + W) = dim U + dim W − dim(U ∩ W).

Direct sum U⊕W: U ∩ W = {0}, dim(U⊕W) = dim U + dim W.