Linear Algebra,
rebuilt for the exam.
Twenty-four sections, in the order you learned them. Each one opens with what you actually need to be able to do, states the definitions and theorems precisely, walks one representative problem step by step, and then hands you concept questions and true/false traps to see whether it stuck.
Matrices and Systems of Equations
Elimination, echelon forms, and the algebra of matrices — the computational engine everything else runs on.
Consistency, equivalent systems, elementary row operations, back substitution.
Gaussian elimination, lead vs. free variables, over/underdetermined and homogeneous systems.
Ax as a linear combination of columns, matrix multiplication, transpose, symmetry.
Algebraic rules, the identity, inverses, and why (AB)⁻¹ = B⁻¹A⁻¹.
Row operations as left multiplication, row equivalence, the nonsingularity theorem, computing A⁻¹.
Determinants
A single scalar that decides singular vs. nonsingular — and how to compute it without dying.
Vector Spaces
The structural layer: subspaces, independence, basis, dimension, and the four subspaces of a matrix.
Closure, the eight axioms, and the standard examples ℝⁿ, ℝᵐˣⁿ, C[a,b], Pₙ.
The subspace test, null space, span, and solution sets as translated subspaces.
The definition that actually gets used, determinant and echelon tests, the Wronskian.
Basis = independent + spanning, why every basis has the same size, extending and paring down.
Coordinate vectors, transition matrices, and the U⁻¹V two-step.
Rank, the consistency theorem, and the Rank–Nullity theorem.
Linear Transformations
Maps that respect the structure, and the matrices that represent them in any pair of bases.
Orthogonality
Length, angle, and projection — leading to least squares, orthonormal bases, and QR.
Dot product, length, angle, Cauchy–Schwarz, scalar and vector projections.
Orthogonal complements, the Fundamental Subspaces theorem, direct sums.
Residuals, the normal equations, projection matrices, best-fit polynomials.
Abstract inner products, norms, Frobenius norm, and norms with no Pythagoras.
Orthonormal bases, Parseval, orthogonal matrices, projection via UUᵀ.
Constructing orthonormal bases and the QR factorization for least squares.
Eigenvalues
Directions a matrix only stretches — the characteristic equation, ODE systems, and diagonalization.
Characteristic polynomial, eigenspaces, complex pairs, trace and determinant, similarity.
Y' = AY, general and particular solutions, complex eigenvalues, higher-order systems.
X⁻¹AX = D, defective matrices, powers of A, and the matrix exponential.