MAT1503
Linear Algebra
Comprehensive Study Summary
Week
Topics Covered
1
1.1 Introduction to Systems of Linear Equations | 1.2 Gaussian Elimination
2
1.3 Matrices and Matrix Operations | 1.4 Inverses; Algebraic Properties
3
1.5 Elementary Matrices and Finding Inverses | 1.6 More on Linear Systems
4
2.1 Determinants by Cofactor Expansion | 2.2 Determinants by Row Reduction
5–6
2.3 Properties of Determinants; Cramer's Rule | 4.1 Real Vector Spaces
7
4.2 Subspaces | 4.3 Spanning Sets | 4.4 Linear Independence
8
4.5 Coordinates and Bases | 4.6 Dimension
9
4.8 Row Space, Column Space, and Null Space | 4.9 Rank and Nullity
Based on: Elementary Linear Algebra, 12th Edition — Howard Anton & Anton Kaul
Chapter 1 · Systems of Linear Equations and
Matrices
1.1 Introduction to Systems of Linear Equations
Linear Equations
Definition
A linear equation in n variables x1, x2, …, xn has the form a1x1 + a2x2 + ■ + anxn = b,
where the coefficients ai and the constant b are real numbers, and the ai's are not all
zero. Variables appear only to the first power — no products, roots, or transcendental
functions of variables are allowed.
• A homogeneous linear equation has b = 0.
• A finite collection of linear equations is called a system of linear equations (linear system).
• A solution is an ordered n-tuple (s1, …, sn) that satisfies every equation in the system simultaneously.
Consistency and Solution Types
Every linear system has exactly one of three possible outcomes:
• No solution — the system is inconsistent (parallel planes/lines with no common point).
• Exactly one solution — the system is consistent (unique intersection point).
• Infinitely many solutions — the system is consistent (lines/planes coincide or share a line).
Augmented Matrix
Definition
The augmented matrix of a linear system encodes all coefficients and constants. For a
system with m equations in n unknowns, it is an m×(n+1) matrix [A | b] where A
contains the coefficients and the last column contains the constants from the right-hand
side.
1.2 Gaussian Elimination
Elementary Row Operations
Three operations preserve the solution set of a linear system:
• Interchange: Swap two rows (Ri ↔ Rj).
• Scaling: Multiply a row by a nonzero constant (kRi → Ri).
• Row addition: Add a multiple of one row to another (Ri + kRj → Ri).
Row Echelon and Reduced Row Echelon Form
Row Echelon
Form (REF)
A matrix is in REF if: (1) Any all-zero rows are at the bottom; (2) In each nonzero row,
the first nonzero entry (the leading entry or pivot) is strictly to the right of the leading
entry of the row above.
RREF
A matrix is in Reduced Row Echelon Form (RREF) if it satisfies all REF conditions,
plus: (3) Each leading entry is 1 (a leading 1); (4) Each column containing a leading 1
has zeros in every other position (both above and below).
Gaussian Elimination Algorithm
Forward phase (Gaussian Elimination): Use row operations to reduce the augmented matrix to REF. Then use
back-substitution to solve. Full phase (Gauss-Jordan Elimination): Reduce further to RREF — the solution can
be read off directly.
Theorem
Every matrix has a unique RREF. Two linear systems are equivalent (have the same
solution set) if and only if their augmented matrices row-reduce to the same RREF.
Free Variables and Parametric Solutions
In the RREF, variables corresponding to columns with a leading 1 are called basic (leading) variables; the
remaining variables are free variables. Free variables can take any value — each choice gives a different
solution, producing infinitely many solutions.
1.3 Matrices and Matrix Operations
Matrix Basics
Definition
An m×n matrix is a rectangular array of mn numbers arranged in m rows and n
columns. The entry in row i and column j is denoted aij. Two matrices are equal iff they
have the same size and all corresponding entries are equal.
Arithmetic Operations
• Addition/Subtraction: Only defined for matrices of the same size; perform entrywise.
• Scalar multiplication: Multiply every entry by the scalar k.
• Matrix multiplication (AB): The (i,j) entry of AB is the dot product of the i-th row of A with the j-th column of
B. Requires the number of columns of A to equal the number of rows of B.
• Transpose (AT): Reflect A across its main diagonal; the (i,j) entry of AT is aji.
Key Properties
Matrix multiplication is associative: (AB)C = A(BC). It is distributive over addition:
A(B+C) = AB + AC. Warning: It is generally NOT commutative — AB ≠ BA in general.
Also (AB)T = BTAT.
1.4 Inverses; Algebraic Properties of Matrices
Matrix Inverse
An n×n matrix A is invertible (nonsingular) if there exists an n×n matrix B such that
AB = BA = In, where In is the n×n identity matrix. We write B = A-1. If no such B exists, A
is singular.
• For 2×2: If A = [[a,b],[c,d]] and det(A) = ad – bc ≠ 0, then A-1 = (1/det(A))[[d,–b],[–c,a]].
• Computing inverse of n×n A: Row-reduce [A | I]. If A is invertible, the result is [I | A-1].
Properties of
Inverses
(A-1)-1 = A. (AB)-1 = B-1A-1. (AT)-1 = (A-1)T. (kA)-1 = (1/k)A-1 for nonzero scalar k.
A system Ax = b with an invertible matrix A has exactly one solution: x = A-1b.
1.5 Elementary Matrices and a Method for Finding Inverses
Elementary Matrix
An elementary matrix is one obtained from the identity matrix In by a single
elementary row operation. Multiplying A on the left by an elementary matrix E performs
that same row operation on A.
Theorem
Every invertible matrix A is expressible as a product of elementary matrices: A =
E1E2■Ek. Conversely, each elementary matrix is invertible, and its inverse is also an
elementary matrix corresponding to the reverse row operation.
Finding A-1 via Row Reduction
Form the n×2n augmented matrix [A | I]. Apply row operations to reduce A to I. If successful, the right half
transforms into A-1. If A cannot be reduced to I, then A is singular (not invertible).
1.6 More on Linear Systems and Invertible Matrices
Theorem 1.6.1
If A is an invertible n×n matrix, then for every n×1 matrix b, the system Ax = b has
exactly one solution, namely x = A-1b.
Invertible Matrix
Theorem (partial)
For an n×n matrix A, the following are equivalent: (a) A is invertible; (b) Ax = 0 has only
the trivial solution; (c) The RREF of A is In; (d) A is expressible as a product of
elementary matrices; (e) Ax = b is consistent for every n×1 matrix b.
For a homogeneous system Ax = 0: it always has the trivial solution x = 0. If A is invertible, this is the ONLY
solution. If A is singular (or non-square with more unknowns than equations), there are infinitely many solutions.
Chapter 2 · Determinants
2.1 Determinants by Cofactor Expansion
Minor & Cofactor
For an n×n matrix A, the minor Mij is the determinant of the (n–1)×(n–1) submatrix
obtained by deleting row i and column j. The cofactor Cij = (–1)i+j Mij. The sign pattern
follows a checkerboard: + where i+j is even, – where i+j is odd.
Determinant
(Cofactor
Expansion)
For an n×n matrix A, the determinant along row i is: det(A) = ai1Ci1 + ai2Ci2 + ■ +
ainCin. The same value is obtained by expanding along any row or any column. Base
case: det([a]) = a for a 1×1 matrix; det([[a,b],[c,d]]) = ad – bc for 2×2.
Strategy: Choose the row or column with the most zeros to minimize computation. For a triangular matrix,
det(A) = product of diagonal entries.
2.2 Evaluating Determinants by Row Reduction
Row reduction provides an efficient method for computing determinants of large matrices.
• Row swap: Interchanging two rows reverses the sign of the determinant.
• Row scaling: Multiplying a row by scalar k multiplies det by k.
• Row addition: Adding a multiple of one row to another does not change the determinant.
Method: Reduce A to upper triangular form U via row operations, keeping track of sign changes and scaling.
Then det(A) = (±1) × (product of diagonal entries of U), adjusted for any row scalings used.
2.3 Properties of Determinants; Cramer's Rule
Key Properties
• det(AT) = det(A) — transposing does not change the determinant.
• det(AB) = det(A) · det(B) — the product rule.
• A is invertible ■ det(A) ≠ 0.
• If A is invertible, det(A-1) = 1/det(A).
• If A has two identical rows (or columns), then det(A) = 0.
• If A has a row (or column) of zeros, then det(A) = 0.
Adjugate and Inverse Formula
Adjugate
The adjugate (classical adjoint) of A, written adj(A), is the transpose of the cofactor
matrix: [adj(A)]ij = Cji. If det(A) ≠ 0, then A-1 = (1/det(A)) · adj(A).
Cramer's Rule
Cramer's Rule
If Ax = b is a linear system of n equations in n unknowns and det(A) ≠ 0, then the
system has a unique solution given by: xi = det(Ai) / det(A), where Ai is the matrix
obtained by replacing the i-th column of A with the column vector b.
Note: Cramer's Rule is primarily of theoretical value. Gaussian elimination is far more efficient for numerical
computation, especially for large systems.
Chapter 4 · General Vector Spaces
4.1 Real Vector Spaces
Vector Space
A vector space V is a nonempty set with two operations — addition (u + v) and scalar
multiplication (ku) — satisfying 10 axioms for all u, v, w in V and scalars k, m: (1)
Closure under addition; (2) Commutativity: u+v = v+u; (3) Associativity of addition; (4)
Zero vector: 0+u = u; (5) Additive inverses: u+(–u) = 0; (6) Closure under scalar mult.;
(7) k(u+v) = ku+kv; (8) (k+m)u = ku+mu; (9) k(mu) = (km)u; (10) 1u = u.
Examples of Vector Spaces
• Rn: n-tuples of real numbers with standard addition and scalar multiplication.
• Mmn: The set of all m×n matrices with matrix addition and scalar multiplication.
• Pn: Polynomials of degree ≤ n; P∞ = all polynomials.
• C[a,b]: Continuous functions on [a,b] — addition and scalar multiplication defined pointwise.
Basic
Consequences
In any vector space: (a) 0u = 0 (zero scalar times any vector = zero vector); (b) k·0 = 0
(any scalar times zero vector = zero vector); (c) (–1)u = –u; (d) if ku = 0 then k = 0 or u
= 0.
4.2 Subspaces
Subspace
A nonempty subset W of a vector space V is a subspace of V if W is itself a vector
space under the same operations. W need only inherit the axioms — it must be verified
separately.
Subspace Test
A nonempty subset W of a vector space V is a subspace of V if and only if: (1) If u and v
are in W, then u+v is in W (closed under addition); (2) If u is in W and k is any scalar,
then ku is in W (closed under scalar multiplication).
Common Subspaces
• The zero subspace {0} is a subspace of every vector space.
• V is always a subspace of itself.
• Any line or plane through the origin in R2 or R3 is a subspace.
• Lines or planes NOT through the origin are not subspaces (fail the zero vector condition).
4.3 Spanning Sets
Linear
Combination &
Span
A linear combination of vectors v1, …, vr in V is any expression k1v1 + k2v2 + ■ + krvr,
where the ki are scalars. The span of {v1, …, vr} is the set of ALL linear combinations of
these vectors, denoted span{v1, …, vr}.
Theorem
If v1, …, vr are vectors in V, then span{v1, …, vr} is a subspace of V. It is the smallest
subspace of V that contains all of v1, …, vr.
We say {v1, …, vr} spans V (or is a spanning set for V) if every vector in V can be written as a linear
combination of v1, …, vr, i.e., span{v1, …, vr} = V.
4.4 Linear Independence
Linear
Independence
Vectors v1, …, vr in a vector space V are linearly independent if the only solution to
k1v1 + ■ + krvr = 0 is the trivial solution k1 = k2 = ■ = kr = 0. They are linearly
dependent if a nontrivial solution (with some ki ≠ 0) exists.
Geometric Interpretation & Key Facts
• In Rn, vectors are linearly dependent iff one of them is a linear combination of the others.
• A set containing the zero vector is always linearly dependent.
• A single nonzero vector is always linearly independent.
• In Rn, any set of more than n vectors is linearly dependent.
Test via Matrices
Vectors v1, …, vr in Rn are linearly independent iff the matrix A = [v1 | v2 | … | vr]
(columns = vectors) has only the trivial solution to Ax = 0, i.e., iff A has rank r.
4.5 Coordinates and Bases
Basis
A set S = {v1, …, vn} of vectors in V is a basis for V if: (1) S spans V, and (2) S is
linearly independent. Equivalently, S is a basis iff every vector in V can be written as a
linear combination of S in exactly one way.
Coordinates
If S = {v1, …, vn} is a basis for V and v = c1v1 + ■ + cnvn, then the scalars c1, …, cn are
the coordinates of v relative to S, written (v)S = (c1, …, cn).
Standard Bases
• Rn: Standard basis e1 = (1,0,…,0), e2 = (0,1,…,0), …, en = (0,…,0,1).
• Pn: Standard basis {1, x, x2, …, xn}.
• M22: Standard basis consists of the four matrices with exactly one 1 and three 0s.
4.6 Dimension
Dimension
The dimension of a vector space V, written dim(V), is the number of vectors in any
basis for V. If V = {0}, we define dim(V) = 0. V is finite-dimensional if it has a finite
basis; otherwise it is infinite-dimensional.
Dimension
Theorem
All bases for a finite-dimensional vector space V have the same number of vectors.
Key Dimension Facts
• dim(Rn) = n. dim(Pn) = n+1. dim(Mmn) = mn.
• If dim(V) = n, then: any n linearly independent vectors form a basis; any n vectors that span V form a basis.
• If W is a subspace of V, then dim(W) ≤ dim(V).
• If dim(W) = dim(V) and W is a subspace of V, then W = V.
Plus/Minus
Theorem
If S is a linearly independent set in V and v ∉ span(S), then S∪{v} is linearly
independent. If S spans V and v ∈ span(S), then removing v from S still spans V.
4.8 Row Space, Column Space, and Null Space
Row Space
The row space of an m×n matrix A is the subspace of Rn spanned by the rows of A.
Column Space
The column space of A (written col(A)) is the subspace of Rm spanned by the columns
of A. col(A) = {Ax : x ∈ Rn} — the set of all vectors that can be written as Ax.
Null Space
The null space (or kernel) of A, written null(A), is the solution set of Ax = 0: null(A) = {x
∈ Rn : Ax = 0}. This is always a subspace of Rn.
Finding These Spaces
• Row space: Row-reduce A; the nonzero rows of any REF of A form a basis.
• Column space: Row-reduce A; columns of the ORIGINAL A corresponding to pivot columns form a basis.
• Null space: Solve Ax = 0 by row reduction; express solution in terms of free variables as vectors to get a
basis.
Consistency
Criterion
The system Ax = b is consistent iff b is in the column space of A.
4.9 Rank, Nullity, and the Fundamental Matrix Spaces
Rank & Nullity
The rank of A, written rank(A), is the dimension of the row space of A (= dimension of
column space of A). The nullity of A, written nullity(A), is the dimension of the null
space of A.
Rank–Nullity
Theorem
(Dimension
Theorem for
Matrices)
For any m×n matrix A: rank(A) + nullity(A) = n (the number of columns). Equivalently:
(number of pivot columns) + (number of free variable columns) = n.
The Four Fundamental Subspaces
Every m×n matrix A has four associated fundamental subspaces:
Subspace
Lies in
Dimension
Row space of A
R^n
rank(A)
Column space of A
R^m
rank(A)
Null space of A
R^n
nullity(A) = n – rank(A)
Null space of A^T
R^m
m – rank(A)
Key Equivalences
involving Rank
For an n×n matrix A: A is invertible ■ rank(A) = n ■ nullity(A) = 0 ■ the rows form a
basis for Rn ■ the columns form a basis for Rn.
Consistency and the Column Space
• Ax = b is consistent iff rank(A) = rank([A | b]).
• If consistent, the solution is unique iff nullity(A) = 0.
• If consistent and nullity(A) = k > 0, then there are infinitely many solutions parameterized by k free variables.
End of MAT1503 Summary — Elementary Linear Algebra (Anton & Kaul, 12th Ed.)