MATH1030
Linear algebra I
Start with systems of equations and row reduction, then connect matrix calculation to span, independence, basis and dimension. Determinants, eigenvectors and inner products reveal the algebraic and geometric structure behind those calculations.
Your progress
0/ 35 sections marked complete
Next section: 1.1 Equations and solution sets
Completion marks are saved in this browser.
This course in the web
In the MATH1030 notes
- Theorem cards
- 140
- Definitions
- 68
- Counterexamples
- 13
- Concept lenses
- 12
- Histories of ideas
- 3
- Proof X-rays
- 5
Sections with a history of an idea
- Gaussian elimination and RREF2.3
- Determinants and cofactor expansion7.1
- Gram-Schmidt orthogonalization9.3
1 of these is on the timeline so far. See the timeline →
2 concept lenses are recorded in the web.
Counts cover the 35 public sections of this course and what is written in them, not recorded connections. 3 of these sections are indexed into the web in detail so far.
In the webTheorem cards 140 · Definitions 68 · Counterexamples 13 · Concept lenses 12 · Histories of ideas 3 · Proof X-rays 5
1 of these is on the timeline so far. See the timeline →
2 concept lenses are recorded in the web.
Counts cover the 35 public sections of this course and what is written in them, not recorded connections. 3 of these sections are indexed into the web in detail so far.
Chapter 1 · 2 sections
Systems of equations
Learn to read equations as full solution sets.
Theorems 5 · Definitions 7Counterexamples 1Lenses 1Proof X-rays 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 1.1
Equations and solution sets
Read a linear system as a collection of conditions and describe its full solution set carefully.
- 1.2
Reading theorems and proof language
Learn how to read definitions, theorem statements, equivalences, uniqueness claims, and counterexamples before the course becomes proof-heavy.
Chapter 2 · 5 sections
Matrices and elimination
Build matrix intuition and use row reduction with purpose.
Theorems 16 · Definitions 8Counterexamples 1Lenses 1History 1Proof X-rays 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 2.1
Matrix basics
Build matrix intuition before you row-reduce: size, entries, rows, columns, and arithmetic meaning.
- 2.2
Augmented matrices and row operations
Translate a system into an augmented matrix and understand what each row operation preserves.
- 2.3
Gaussian elimination and RREF
See Gaussian elimination as a sequence of purposeful moves, not just memorized mechanics.
- 2.4
Solution-set types
Classify whether a system has one solution, infinitely many solutions, or no solution by reading its reduced form.
- 2.5
Existence of row-echelon forms
Read the optional induction proof that every matrix is row-equivalent to an REF and then to an RREF.
Chapter 3 · 6 sections
Matrix algebra
Matrix arithmetic, multiplication, transpose, special and elementary matrices, and block structure.
Theorems 18 · Definitions 13Counterexamples 1Lenses 2
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 3.1
Matrix addition, subtraction, and scalar multiplication
Learn which matrix operations are entrywise, why matching sizes matter, and how the zero matrix behaves like the additive identity.
- 3.2
Matrix multiplication, identity matrices, and linear systems
Learn when matrix products are defined, how the row-by-column rule works, and why the identity matrix matters for solving linear systems.
- 3.3
Transpose, symmetric, and skew-symmetric matrices
Use transpose, symmetry, skew-symmetry, and commuting products to read matrix structure rather than treating formulas as isolated tricks.
- 3.4
Special matrices
Meet diagonal, triangular, identity, and elementary matrices, and see why their special shapes make later arguments shorter.
- 3.5
Elementary row-operation matrices
Represent elementary row operations as left multiplication by elementary matrices, then use reverse row operations to understand invertibility.
- 3.6
Block matrices
Partition a large matrix into smaller pieces so addition and multiplication can be carried out block by block without losing meaning.
Chapter 4 · 2 sections
Solution structure
Homogeneous systems, null spaces, and the shape of full solution sets.
Theorems 9 · Definitions 7Lenses 1Proof X-rays 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 4.1
Homogeneous systems and null space
Study homogeneous systems carefully, then use null spaces to describe every solution as a structured set rather than a loose list of examples.
- 4.2
Set language and solution sets
Use set notation, membership, solution sets, null spaces, spans, and set equality carefully in linear algebra arguments.
Chapter 5 · 2 sections
Invertibility
Understand when a matrix can be undone and why that matters.
Theorems 10 · Definitions 3Proof X-rays 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 5.1
Invertible matrices
Connect inverse matrices, row reduction, and the practical meaning of nonsingularity.
- 5.2
RREF uniqueness and well-defined rank
Prove uniqueness of reduced row-echelon form and use it to make rank independent of the chosen row-reduction path.
Chapter 6 · 8 sections
Vector spaces
Move from matrix procedures to the structure of spaces, span, independence, and basis.
Theorems 27 · Definitions 12Counterexamples 4Lenses 3Proof X-rays 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 6.1
Vector spaces
Start from familiar examples and learn what the vector-space axioms are trying to protect.
- 6.2
Subspaces
Use the subspace test to separate genuine linear structure from lookalikes that fail closure or miss the zero vector.
- 6.3
Linear combinations and span
Treat linear combinations as controlled building instructions, then see span as every vector you can build that way.
- 6.4
Linear dependence and independence
Read dependence as redundancy, and independence as the point where every coefficient truly matters.
- 6.5
Basis and dimension
See why a basis is the smallest complete coordinate system for a space, and why dimension counts how many directions are really needed.
- 6.6
Column space, row space, and rank
Use row reduction and basis ideas together to read column space, row space, and rank without confusing what row operations actually preserve.
- 6.7
Matrix subspaces, basis, and dimension
Extend span, basis, and dimension from column-vector subspaces to matrix subspaces such as all matrices, upper triangular matrices, and skew-symmetric matrices.
- 6.8
Basis extension and change of basis
Use basis existence, the replacement theorem, and change-of-basis matrices to compare coordinate systems rigorously.
Chapter 7 · 3 sections
Determinants
Determinants, cofactor formulas, and the structural algebra that connects row operations, transpose, and invertibility.
Theorems 16 · Definitions 4Counterexamples 2Lenses 2History 1
Chapter outline →See this chapter on the web →3 of 3 sections indexed
- 7.1
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.
- 7.2
Row operations, products, and invertibility
Track exactly how row operations change determinants, then connect that behavior to multiplicativity, inverse matrices, and invertibility tests.
- 7.3
Transpose, column operations, and Cramer's rule
Use transpose and column operations to read determinants from a second angle, then finish the chapter with adjoints, inverse formulas, and Cramer's rule.
Chapter 8 · 3 sections
Eigenvalues and diagonalization
Eigenvalues, eigenspaces, similarity, and diagonalization as the next structural layer after determinants.
Theorems 21 · Definitions 6Counterexamples 2Lenses 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 8.1
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.
- 8.2
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.
- 8.3
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 · 4 sections
Inner products and orthogonality
Inner products, orthogonality, orthonormal bases, and Gram-Schmidt as the geometric layer after eigenvalues.
Theorems 18 · Definitions 8Counterexamples 2Lenses 1History 1
Chapter outline →See this chapter on the web →Not yet indexed in the web
- 9.1
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.
- 9.2
Orthogonal sets and orthonormal bases
Use orthogonality to build orthogonal and orthonormal bases, then read coefficients without solving a linear system every time.
- 9.3
Gram-Schmidt orthogonalization
Apply Gram-Schmidt to turn a basis into an orthogonal or orthonormal basis while preserving the same span.
- 9.4
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.