In This Lesson Linear Transformations Vector Spaces & Subspaces Basis & Dimension 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 Rank & Nullity 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 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 Projection: Onto a subspace — key in least-squares regression 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).
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 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 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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
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).