Vector Spaces & Linear Transformations

Abstract vector spaces unify vectors, polynomials, signals and functions. Learn vector spaces and subspaces, span, linear independence, basis and dimension, column space, null space and rank–nullity, linear transformations with their matrices, kernel and range, and change of basis.

The same ideas — combinations, independence, bases — apply to 3D vectors, audio signals (Fourier basis), polynomials (shape functions in FEM) and images (wavelet bases). Recognising structure lets engineers choose bases that make problems simple.

An analogy

A basis is like a set of LEGO brick types: every model (vector) in the space can be built from them in exactly one way, and none of the brick types can be built from the others.

Span and independence

Vectors are independent if none is a combination of the others. Test: put them as columns and row-reduce — a pivot in every column means independent.

Subspaces

Subsets closed under addition and scaling that contain 0: lines/planes through the origin, solution sets of Ax = 0, polynomials of degree ≤ n.

Fundamental subspaces

Column space (reachable outputs b), null space (inputs mapped to 0 — mechanism modes, rigid-body motions), row space and left null space. Rank–nullity links them.

Transformations and bases

Every linear map between finite-dimensional spaces is a matrix once bases are chosen. A good basis (eigenvectors, Fourier modes) diagonalises the problem.

Formulas

• Linear combination: c₁v₁ + c₂v₂ + … + cₖvₖ
• Independence: c₁v₁ + … + cₖvₖ = 0 ⇒ all cᵢ = 0
• Rank–nullity: rank A + dim N(A) = n (columns)
• Linearity: T(au + bv) = aT(u) + bT(v)
• Change of basis: [x]_B′ = P⁻¹[x]_B; A′ = P⁻¹AP

Worked example: Independence test

Given: (1, 2, 3), (2, 4, 6), (1, 0, 1)

Find: Independent?

• Second = 2 × first
• rank = 2

Answer: Dependent; they span a plane (dimension 2)

Worked example: Null space

Given: A = [[1, 2, 3], [2, 4, 6]]

Find: N(A) dimension.

• rank = 1, n = 3
• dim N = 2 (x = −2s − 3t, y = s, z = t)

Answer: 2-dimensional null space

Common mistakes

• Wrong: Thinking any set spanning a space is a basis. — Right: A basis must also be independent.
• Wrong: Subspaces not containing 0. — Right: Every subspace contains the zero vector.

Applications

• Structural mechanisms: Null space = rigid-body/mechanism modes.
• Signal processing: Fourier and wavelet bases.
• FEM: Shape functions as a basis.

Key takeaways

• Span, independence, basis, dimension.
• Column and null spaces.
• rank + nullity = n.
• Choose bases that simplify problems.

Going further

• Dual spaces: Linear functionals.
• Infinite-dimensional spaces: Function spaces (functional analysis).
• Quotient spaces: Removing rigid-body modes.