In This Lesson Linear Transformations Vector Spaces & Subspaces Basis & Dimension Rank & Nullity A linear transformation T: V → W satisfies:
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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:
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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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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).
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