In This Lesson Linear Transformations Vector Spaces & Subspaces Basis & Dimension Rank & Nullity 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 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:
Rotation by θ: Uses sine and cosine in the matrix Reflection: Across a line or plane Scaling: Diagonal matrix with scale factors Projection: Onto a subspace — key in least-squares regression 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 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:
Polynomials of degree ≤ n — connects to polynomial algebra 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 Continuous functions on [a, b] — connects to continuity in calculus Solutions to homogeneous DEs — connects to differential equations 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 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
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 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
rank(A) = dim(Col A) = dim(Row A)
nullity(A) = dim(Nul A)
Rank-Nullity Theorem: rank(A) + nullity(A) = n
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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).