MATH1030: Linear algebra I
Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.
Use the sidebar to move chapter by chapter, or jump directly into a section below.
Chapter 1
Systems of equations
Learn to read equations as full solution sets.
1.1 Equations and solution sets
Read a linear system as a collection of conditions and describe its full solution set carefully.
Chapter 2
Matrices and elimination
Build matrix intuition and use row reduction with purpose.
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.
Chapter 3
Matrix algebra
Matrix multiplication, transpose, and structural matrix notation.
3.1 Matrix multiplication and identity matrices
Learn when matrix products are defined, how the row-by-column rule works, and why the identity matrix matters for solving linear systems.
3.2 Transpose and special matrices
Use transpose, symmetry, commuting products, and block notation to read matrix structure rather than treating formulas as isolated tricks.
Chapter 4
Solution structure
Homogeneous systems, null spaces, and the shape of full solution sets.
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.
Chapter 5
Invertibility
Understand when a matrix can be undone and why that matters.
5.1 Invertible matrices
Connect inverse matrices, row reduction, and the practical meaning of nonsingularity.
Chapter 6
Vector spaces
Move from matrix procedures to the structure of spaces, span, independence, and basis.
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.
Note collections
MATH1030: Linear algebra I
Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.
Chapter 1
Systems of equations
Learn to read equations as full solution sets.
Chapter 2
Matrices and elimination
Build matrix intuition and use row reduction with purpose.
Chapter 3
Matrix algebra
Matrix multiplication, transpose, and structural matrix notation.
Chapter 4
Solution structure
Homogeneous systems, null spaces, and the shape of full solution sets.
Chapter 5
Invertibility
Understand when a matrix can be undone and why that matters.
Chapter 6
Vector spaces
Move from matrix procedures to the structure of spaces, span, independence, and basis.