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linear algebra notes

Handwritten course notes, key theorems, computational algorithms, and worked problem sets covering linear systems, transformations, and vector spaces.

Available Notes 9 documents

Sections 1.3 & 1.4 PNG Image • 1.3 MB

1.3 & 1.4 — Vector Equations & Matrix Equations Ax = b

Vectors in ℝⁿ, linear combinations, Span{v₁...vₚ}, and matrix equations Ax = b with existence/uniqueness theorems.

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Section 1.5 PNG Image • 1.4 MB

1.5 — Solution Sets of Linear Systems

Homogeneous vs. non-homogeneous systems (Ax = 0, Ax = b), parametric vector forms, and 4-step solution method.

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Section 1.7 PNG Image • 914.2 KB

1.7 — Linear Independence

Linear dependence and independence, trivial solutions, free variables, and geometric conditions.

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Section 1.8 PNG Image • 1.7 MB

1.8 — Matrix Transformations

Linear transformations T: ℝⁿ → ℝᵐ, domain/codomain/range, superposition principle, and matrix operations.

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Section 1.9 PNG Image • 673.0 KB

1.9 — The Matrix of a Linear Transformation

Standard matrix representation [T(e₁)...T(eₙ)], onto (surjective) and one-to-one (injective) mappings.

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Section 2.1 PNG Image • 211.2 KB

2.1 — Matrix Operations

Matrix multiplication rules and dimensions, algebraic properties, non-commutativity, transpose Aᵀ, and matrix powers.

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Section 2.2 PNG Image • 237.1 KB

2.2 — Invertible Matrices

Matrix invertibility conditions, row-reduction algorithm [A | Iₙ] ~ [Iₙ | A⁻¹], elementary matrices, and algebraic inverse rules.

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Section 2.3 PNG Image • 196.6 KB

2.3 — The Invertible Matrix Theorem

The 12 equivalent statements of the Invertible Matrix Theorem (IMT): invertibility, trivial kernel, full column span, linear independence, n pivots, and one-to-one maps.

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Section 2.5 PNG Image • 208.4 KB

2.5 — LU Factorization

Lower & upper triangular matrix decomposition A = LU, construction via row operations without scaling or row swaps, and two-step solving (Ly = b, Ux = y).

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