In This Lesson 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 Linear Transformations Vector Spaces & Subspaces 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 Basis & Dimension 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 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)
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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:
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).