MATH1550: Methods of Matrices and Linear Algebra
Matrix methods, real vector spaces, spectral structure and rank-aware approximation.
Equations and solution sets
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Chapter 1
Linear systems
Linear systems
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Equations and solution sets
Treat a linear system as a set of simultaneous conditions, classify its solution set carefully, and see why equation operations preserve that set.
Chapter 2
Matrices and elimination
Matrices and elimination
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Matrix basics
Read matrices carefully as mathematical objects: size, entries, equality, basic operations, and how systems of equations become matrix statements.
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Augmented matrices and row operations
Translate a system into `Ax = b`, package it as `[A|b]`, and understand exactly why elementary row operations preserve the solution set.
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Gaussian elimination and RREF
Gaussian elimination preserves a system's solution set while exposing its pivot equations and free variables.
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Solution-set types
Use RREF to classify a linear system rigorously, identify pivot and free variables, and explain why a system has exactly one solution, infinitely many solutions, or no solution.
Chapter 3
Matrix algebra
Matrix algebra
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Matrix multiplication, identity matrices, and linear systems
Learn when matrix products are defined, how the row-by-column rule and identity matrices work, and how multiplication encodes linear systems.
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Transpose, symmetric, and skew-symmetric matrices
Use transpose, symmetry, and commuting products to read matrix structure and algebraic behavior.
Chapter 4
Solution structure
Solution structure
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Homogeneous systems and null space
The null space describes the differences between solutions, their free coordinates, and the structure of every consistent solution set.
Chapter 5
Invertibility and rank
Invertibility and rank
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RREF uniqueness and well-defined rank
Prove that every row-equivalence class has only one reduced row-echelon form, then use that theorem to make rank independent of the chosen reduction path.
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Invertible matrices
Invertibility connects reversible matrix actions, unique solutions for every right-hand side, and the algebra of inverse matrices.
Chapter 6
Spaces and bases
Spaces and bases
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Vector spaces
Define vector spaces carefully, compare the standard examples, and prove the first structural facts that every later span, basis, and dimension argument depends on.
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Subspaces
Use the subspace test carefully, prove the first structural consequences, and learn the standard matrix and equation-defined examples that recur throughout linear algebra.
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Linear combinations and span
Treat span as a precise set of outputs, connect it to linear systems, and learn how to decide whether a vector can or cannot be built from a given list.
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Linear dependence and independence
Detect redundancy in a list of vectors, test independence by direct relations and row reduction, and read the geometric meaning in low dimensions.
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Basis and dimension
A basis gives unique coordinates; dimension records the common size of all bases and provides criteria for spanning and independence.
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Column space, row space, and rank
Read a matrix as both a source of outputs and a source of linear relations, then use rank to count the genuinely independent directions that remain.
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Basis extension and change of basis
Turn the basis theorems into usable tools: extend independent lists, replace old basis vectors, and compare coordinates across two bases.
Chapter 7
Determinants
Determinants
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Determinants and cofactor expansion
Define determinants carefully through minors and cofactors, then learn how cofactor expansion turns one scalar into a precise summary of square-matrix structure.
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Row operations, products, and invertibility
Track exactly how row operations change determinants, then connect that behavior to multiplicativity, inverse matrices, and invertibility tests.
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Transpose, column operations, and Cramer's rule
Transpose transfers determinant identities from rows to columns; cofactor cancellation then gives the adjoint formula and Cramer’s rule.
Chapter 8
Eigenvalues and diagonalization
Eigenvalues and diagonalization
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Eigenvalues, eigenvectors, and eigenspaces
Define eigenvalues through the equation Av=λv, then recast the same idea as a null-space and determinant question so the structure becomes computable.
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Diagonalization and similarity
Treat diagonalization as a basis change built from eigenvectors, then use similarity to explain when a matrix can be simplified without changing its essential eigenvalue data.
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Characteristic polynomials and diagonalization tests
Use characteristic polynomials, algebraic and geometric multiplicity, and the distinct-eigenvalue test to decide when eigenvalue data is enough for diagonalization.
Chapter 9
Orthogonality
Orthogonality
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Inner products, norms, and angles
Define the standard inner product and norm on R^m, then connect those formulas to length, angle, and the first structural inequalities.
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Orthogonal sets and orthonormal bases
Use orthogonality to build orthogonal and orthonormal bases, then read coefficients without solving a linear system every time.
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Gram-Schmidt orthogonalization
Apply Gram-Schmidt to turn a basis into an orthogonal or orthonormal basis while preserving the same span.
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Cauchy-Schwarz and triangle inequalities
Study Cauchy-Schwarz and triangle inequalities as the two core estimates that control length, angle, and equality cases in inner-product spaces.
Chapter 10
Complete spectral methods
Complete spectral methods
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Repeated Eigenvalues and Complete Diagonalization
Repeated Eigenvalues and Complete Diagonalization
Chapter 11
Rank-aware approximation
Rank-aware approximation
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Rank-Aware Gram–Schmidt, QR, and Least Squares
Rank-Aware Gram–Schmidt, QR, and Least Squares