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 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 Moqklhtp4CDMBXXiMwR2bYoCIXpefdW0ADJJcJ454M/vhNQ9hekMHPRzgSDoRzv4V6UR8ZM70mDA/yrFd1QqIlmpYqIE2UN62uY4qG7HATPLKm8AoqAEbIQ0+AAAopQlduQmqbUkfo1nf2epeIeLK0rUcJ0ynNj2QuUQQGlCpABivokYxLP6o+Jubqyo3hjaaXLa7V2lavlgnYzC2ZhNq87ZXGpIvsqypPh8dMYNmZguJhUAaC9Jus7eb/j6rtdwdXl1PxyOZ84uFwkxoenmHl3wpaHDovJ69t1w1wdfgz+RucE6TIB5KZdblVyV3abQr+6Uqx2Qlv1l/SOXt4Y/nbAIWpMz9OmphzvT5j/dDsmHaibWD+RO/pWFZ8NdgEfAzVVDGHJ+dUDDjw1ACwHWTM634WwRhUurKjUh4kJa12/ZAQ6y2u4oayRgZcveBaWd04CT3jMcIm3i7Euerx/4oOlT71XbdJ2wjYtvSWyCF+E9ppCsJGETGeBckWG1yjve25/hMrWkGKDQC/pe/egTC+3kDJZa3Y9ldkCKRE2dasdr9h60IEdZmkzNfFnL/r+BQ/+VsbIprjcOu4sWWweYL7YsVUADbVJeVSPWtTg2TlXTZ8nfC4L5cbz8BmtBdEXUWo 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
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).