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Transformations & Vector Spaces

Understand the abstract structures that unify linear algebra.

Linear Transformations

A linear transformation T: V → W satisfies:

T(u + v) = T(u) + T(v)  (additivity)
T(cv) = cT(v)  (homogeneity)

Every linear transformation from ℝⁿ to ℝᵐ can be represented as multiplication by an m×n matrix. Geometric examples:

These connect directly to coordinate geometry transformations.

Vector Spaces & Subspaces

A vector space V over ℝ is a set with addition and scalar multiplication satisfying 8 axioms (closure, associativity, commutativity, identity, inverse, compatibility, distributivity). Examples beyond ℝⁿ:

A subspace is a subset closed under addition and scalar multiplication. Important subspaces of a matrix A: column space (Col A), row space, null space (Nul A).

Basis & Dimension

A basis is a linearly independent spanning set.
dim(V) = number of vectors in any basis
dim(ℝⁿ) = n, with standard basis e₁, e₂, …, eₙ

Change of basis transforms coordinates between different bases — essential when working with eigenvectors as a basis (diagonalization).

Rank & Nullity

rank(A) = dim(Col A) = dim(Row A)
nullity(A) = dim(Nul A)
Rank-Nullity Theorem: rank(A) + nullity(A) = n
The Rank-Nullity Theorem is a dimension-counting result: the "input space" ℝⁿ splits into the part that maps to nonzero outputs (rank) and the part that maps to zero (nullity). This connects to solution counts for systems of equations: unique (full rank), infinite (nullity > 0), or none (inconsistent).

Related Topics

Vectors & MatricesConcrete objects before abstracting to spaces. Coordinate GeometryTransformations in the coordinate plane are linear maps. Second-Order DEsSolution spaces of linear DEs are vector spaces. Unit CircleRotation matrices use trigonometric functions.