A First Course in Linear Algebra PDF Download – Beezer (BS Math/BSCS)

BS Math and BSCS students can download the complete linear algebra textbook “A First Course in Linear Algebra” by Robert A. Beezer, Professor of Mathematics at the University of Puget Sound. The book is free from its official source and licensed under the GNU Free Documentation License, currently at Version 3.50. It covers a full first course in linear algebra — systems of linear equations, vectors, matrices, vector spaces, determinants, eigenvalues, linear transformations, and representations — building every result from precise definitions and complete proofs rather than skipping straight to computation.

What sets this book apart for exam preparation is its 24 fully worked “Archetypes,” the same handful of example systems, vectors and matrices carried through every new technique in the book, so a student can check a fresh computation against a system they already understand. Combined with three Reading Questions at the start of every section and over 500 exercises coded by type — computational, mixed, or theoretical — the book works equally well for a computational first course and for a proof-based bridge course, which is exactly the range BS Math and BSCS linear algebra courses in Pakistan need to cover.

Book Overview

Chapter List

Chapter 1: Systems of Linear Equations

Difficulty: Easy · Key topics: linear equations, row reduction, reduced row-echelon form, consistent and inconsistent systems, free variables, homogeneous systems, nonsingular matrices

The book opens by asking what linear algebra actually studies, then gets to work on the practical problem of solving systems of linear equations. Equation operations are proved to preserve the solution set, which justifies row-reducing the augmented matrix of coefficients to a unique reduced row-echelon form. From that form, one theorem tells you immediately whether a system is consistent, and if it is, whether it has one solution or infinitely many, governed by the number of free variables. Homogeneous systems get their own section because they are always consistent, and the chapter closes by defining a nonsingular matrix as one whose associated homogeneous system has only the trivial solution — a property the rest of the book keeps testing in new ways.

Key Points:

• A system is consistent when it has at least one solution; reading its reduced row-echelon form tells you consistency immediately.
• Row reduction uses only three equation operations, and the book proves they never change a system’s solution set.
• Every matrix has one and only one reduced row-echelon form — the result is unique no matter which sequence of row operations you use to reach it.
• A consistent system with more variables than equations always has infinitely many solutions.
• A homogeneous system is always consistent, since setting every variable to zero is always a solution.
• A matrix is nonsingular exactly when its associated homogeneous system has only the trivial (all-zero) solution.

Memory Tip: Pivot columns and free variables are opposites — count the leading 1’s in the reduced row-echelon form, and every other column is a free variable. That one count tells you consistency, uniqueness, and the size of the solution set in one look.

Common Mistake: Confusing “no solution” with “infinitely many solutions.” A row that reads 0 = (nonzero number) means no solution; a free variable column with no such row means infinitely many.

Important Questions:

• How can you tell from a matrix’s reduced row-echelon form whether its system is consistent? The system is inconsistent exactly when a row reads 0 = c for some nonzero constant c; otherwise it is consistent.
• Why is a homogeneous system always consistent? Because setting every variable to zero always satisfies every equation, so the all-zero vector is always at least one solution.

Chapter 2: Vectors

Difficulty: Easy · Key topics: vector addition and scalar multiplication, linear combinations, span, linear independence, linear dependence, inner product and orthogonality

Chapter 2 shifts from equations to the objects that make them up: column vectors in complex n-space. Vector addition and scalar multiplication are defined entrywise, and ten familiar properties — commutativity, associativity, distributivity and so on — are proved once and cited freely afterward. Linear combinations and span follow immediately, and one of the chapter’s theorems rewrites everything from Chapter 1 in vector language: the solution set of a consistent system is one particular solution plus every linear combination of a spanning set for the null space. Linear independence is defined as the trivial relation of dependence being the only one, and the chapter proves that a nonsingular matrix’s columns are always linearly independent. A short section on orthogonality, using the inner product and norm over the complex numbers, closes the chapter.

Key Points:

• Column vector addition and scalar multiplication are entrywise, and complex n-space satisfies the ten vector space properties.
• A linear combination is a sum of vectors each scaled by a coefficient; the span of a set is every linear combination its vectors can form.
• A set of vectors is linearly independent exactly when the only linear combination that equals the zero vector uses all-zero scalars.
• Every solution to a consistent system is one particular solution plus a linear combination from the null space of the coefficient matrix.
• A matrix is nonsingular exactly when its columns are linearly independent.
• The inner product of two complex vectors conjugates one of them, which is what keeps a vector’s inner product with itself a nonnegative real number.

Memory Tip: “Particular plus homogeneous” — picture every solution set of a consistent system as one anchor vector plus the whole null space riding on top of it, a point plus a line or plane through the origin.

Common Mistake: Forgetting to conjugate one of the two vectors when computing a complex inner product, which can produce a negative or complex “length” instead of a proper norm.

Important Questions:

• What does it mean for a set of vectors to be linearly independent? That the only linear combination of them equal to the zero vector is the one using all-zero scalars; any other combination reaching zero makes the set dependent.
• How do you describe every solution of a consistent linear system using vectors? As one particular solution plus every linear combination of a spanning set for the null space of the coefficient matrix.

Chapter 3: Matrices

Difficulty: Medium · Key topics: matrix addition, transpose, adjoint, Hermitian matrices, matrix-vector product, matrix multiplication, matrix inverses, the four fundamental subsets

Having handled vectors, the book turns matrices into full algebraic objects in their own right, starting from a formal definition of the set of all m×n matrices. Matrix addition, scalar multiplication, transpose and the complex conjugate are each shown to interact predictably, and the adjoint (conjugate transpose) is introduced for later use with Hermitian matrices. Matrix multiplication is defined through the matrix-vector product and connected straight back to systems of equations: a matrix equation Ax = b is just another way to write a linear system. Matrix inverses follow, tied directly to nonsingularity, and the chapter finishes by naming the four fundamental subsets associated with a matrix — column space, row space, null space and left null space — that later chapters use constantly.

Key Points:

• Matrix addition and scalar multiplication are entrywise; the set of all m×n matrices forms its own vector space with its own zero matrix.
• The transpose swaps rows and columns; a symmetric matrix equals its own transpose.
• The adjoint conjugates every entry and then transposes; a Hermitian matrix equals its own adjoint.
• The matrix-vector product Ax packages an entire system of linear equations as a single matrix equation.
• A square matrix has an inverse exactly when it is nonsingular, and the inverse turns Ax = b into x = A-1b.
• Every matrix has four fundamental subsets attached to it: column space, row space, null space and left null space.

Memory Tip: “Ax = b is Chapter 1 in one symbol” — every system-of-equations fact from Chapter 1 restates as a fact about the matrix-vector product here.

Common Mistake: Assuming matrix multiplication is commutative, so AB = BA. In general it is not, and the order in every computation and proof matters.

Important Questions:

• What condition on a square matrix guarantees it has an inverse? The matrix must be nonsingular — equivalently, its associated homogeneous system has only the trivial solution, or its columns are linearly independent.
• What are the four fundamental subsets associated with a matrix? Its column space, row space, null space, and left null space.

Chapter 4: Vector Spaces

Difficulty: Hard · Key topics: vector space axioms, subspaces, spanning sets, linear independence, bases, dimension, rank and nullity

This is the pivot of the book, where linear algebra becomes properly abstract. A vector space is defined by ten axioms — the same list already proved for column vectors and matrices — so anything satisfying them, whether it looks like a vector or not, gets the whole theory for free. Subspaces are subsets that are themselves vector spaces, and one theorem cuts the ten axioms down to two checks: closure under addition and closure under scalar multiplication. Span, linear independence and spanning sets reappear in this general setting, building to the chapter’s two central definitions: a basis is a linearly independent spanning set, and the dimension of a vector space is the number of vectors in any one of its bases. The chapter proves every basis of a given space has the same size, which is what makes dimension well defined, and closes with properties of dimension used to compare vector spaces for the rest of the book.

Key Points:

• A vector space is any set with addition and scalar multiplication satisfying ten axioms; column vectors and matrices are two examples the book has already proved.
• The subspace test reduces to two conditions: closed under vector addition, and closed under scalar multiplication.
• A basis is a linearly independent set that spans the vector space.
• The dimension of a vector space is the number of vectors in any basis, and this number never changes no matter which basis you pick.
• The null space, column space, row space and the span of any set are each proved to be subspaces in their own right.
• Rank (the dimension of the column space) and nullity (the dimension of the null space) of a matrix are linked by a rank–nullity relationship developed in this chapter.

Memory Tip: Ten axioms, two shortcuts — you almost never re-check all ten; nearly every subspace proof in this book is “closed under addition, closed under scalar multiplication,” full stop.

Common Mistake: Trying to verify all ten vector space axioms every time a subspace comes up, instead of using the two-condition subspace test the chapter actually licenses.

Important Questions:

• What two conditions does the subspace test require to prove a subset is a subspace? That the subset is closed under vector addition and closed under scalar multiplication — the other eight vector space axioms are inherited automatically.
• Why is the dimension of a vector space well defined? Because every basis of the same vector space is proved to have exactly the same number of vectors, so “the number of vectors in a basis” does not depend on which basis you pick.

Chapter 5: Determinants

Difficulty: Medium · Key topics: elementary matrices, expansion about a row or column, determinant of the transpose, row/column swaps, equal rows, the determinant test for nonsingularity

The determinant takes a square matrix and returns a single scalar, and the chapter is upfront that this is unlike anything studied so far — not an algebraic structure, just a very useful function. It is built up carefully: elementary matrices are shown to perform row operations by left multiplication and are always nonsingular, and every nonsingular matrix turns out to be a product of elementary matrices. The determinant itself starts from the 2×2 case, then is defined recursively by expansion about any row or column — proving that every choice of row or column gives the same answer is one of the chapter’s real achievements. The bulk of the chapter is then a catalogue of properties: a zero row or column forces a zero determinant, swapping two rows flips the sign, a repeated row forces a zero determinant, and scaling or combining rows changes the determinant in predictable, provable ways.

Key Points:

• The determinant is computed by expansion about any row or column — every choice gives the same value.
• A matrix with a zero row, a zero column, or two equal rows or columns always has determinant zero.
• Swapping two rows (or two columns) of a matrix multiplies its determinant by −1.
• Elementary matrices perform row operations by left multiplication and are always nonsingular; every nonsingular matrix is a product of elementary matrices.
• A matrix and its transpose always have the same determinant.
• A square matrix is nonsingular exactly when its determinant is nonzero.

Memory Tip: Determinants trade with row operations at a fixed rate — swap two rows and the sign flips, scale a row by k and the determinant scales by k, add a multiple of one row to another and the determinant does not change at all. Learn that table once instead of re-deriving it.

Common Mistake: Reaching for a memorised 2×2 or 3×3 shortcut (like the diagonal trick) and trying to force it onto a 4×4 or larger matrix, where it simply does not apply.

Important Questions:

• What happens to a matrix’s determinant when two of its rows are swapped? The determinant changes sign — it is multiplied by −1 — while its magnitude stays the same.
• How does the determinant test whether a square matrix is nonsingular? A square matrix is nonsingular exactly when its determinant is nonzero; a zero determinant means the matrix is singular.

Chapter 6: Eigenvalues

Difficulty: Hard · Key topics: eigenvalues and eigenvectors, characteristic polynomial, eigenspace, algebraic and geometric multiplicity, similar matrices, diagonalization

A square matrix acts on vectors by matrix-vector multiplication, and for most vectors the result points in a completely new direction — but for a special few, multiplying by A is the same as multiplying by a single scalar. This chapter defines those scalars as eigenvalues and the vectors as eigenvectors, ties eigenvalues to the roots of the characteristic polynomial, and shows each eigenspace is really the null space of a shifted matrix, so all the null-space machinery from earlier chapters applies directly. Because roots of a polynomial can be complex, the chapter is the real payoff for carrying complex numbers throughout the book. Algebraic and geometric multiplicity are distinguished carefully, similar matrices are defined, and the chapter builds toward diagonalization: when a matrix has enough linearly independent eigenvectors, it is similar to a diagonal matrix, which makes computations like high matrix powers dramatically easier.

Key Points:

• An eigenvalue and eigenvector satisfy Ax = λx for a nonzero vector x; every square matrix has at least one eigenvalue, possibly complex.
• Eigenvalues are exactly the roots of the characteristic polynomial, det(A − λI) = 0.
• The eigenspace for a given eigenvalue is the null space of A − λI, so it is automatically a subspace.
• Eigenvectors corresponding to distinct eigenvalues are always linearly independent of each other.
• A singular matrix always has 0 as one of its eigenvalues, linking this chapter back to nonsingularity.
• A matrix is diagonalizable when it has enough linearly independent eigenvectors to form a basis — geometric multiplicity must match algebraic multiplicity for every eigenvalue.

Memory Tip: Read every “eigen” fact back from Ax = λx: a zero eigenvalue means A is singular, distinct eigenvalues give independent eigenvectors for free, and diagonalization is just choosing a basis made entirely of eigenvectors.

Common Mistake: Assuming a matrix is diagonalizable just because its characteristic polynomial has enough roots, without checking that algebraic and geometric multiplicity actually match for every repeated eigenvalue.

Important Questions:

• How do you find the eigenvalues of a matrix A? Solve the characteristic equation det(A − λI) = 0; its roots, which may be complex, are the eigenvalues of A.
• What does it mean for a matrix to be diagonalizable? That it is similar to a diagonal matrix — equivalently, it has a full basis’s worth of linearly independent eigenvectors.

Chapter 7: Linear Transformations

Difficulty: Hard · Key topics: linear transformations, matrix of a transformation, injective and surjective transformations, kernel, range, invertible transformations, isomorphic vector spaces

The chapter frames linear algebra as the study of two objects together — vector spaces and the functions between them that respect their structure. A linear transformation is a function that preserves both addition and scalar multiplication, and one of the chapter’s first theorems shows that any linear transformation between finite-dimensional spaces can be represented by a matrix once a basis is fixed — a bridge that turns questions about abstract functions back into matrix computations. Injective and surjective transformations get their own sections: injectivity is characterized by a trivial kernel, surjectivity by a range that fills the entire codomain. The chapter ends with invertible linear transformations and isomorphic vector spaces — two spaces connected by an invertible linear transformation are, for every algebraic purpose, indistinguishable.

Key Points:

• A linear transformation T satisfies T(u+v) = T(u)+T(v) and T(αu) = αT(u) for all vectors and scalars.
• Every linear transformation between finite-dimensional vector spaces can be represented by a matrix once bases for the domain and codomain are chosen.
• A linear transformation is injective exactly when its kernel contains only the zero vector.
• A linear transformation is surjective exactly when its range equals the entire codomain.
• The set of all linear transformations between two fixed vector spaces is itself a vector space.
• Two vector spaces are isomorphic when an invertible linear transformation connects them; isomorphic spaces share every algebraic property that matters in this course.

Memory Tip: “Injective = trivial kernel, surjective = full range” — the whole of the injective and surjective sections compresses into that one line.

Common Mistake: Checking T(u+v) = T(u)+T(v) and T(αu) = αT(u) on one or two example vectors and calling that a proof of linearity — the definition requires it for every vector and scalar, which needs a general argument, not a numerical check.

Important Questions:

• What two conditions must a function satisfy to be a linear transformation? It must preserve vector addition and scalar multiplication for all vectors and all scalars — not just for a few chosen examples.
• How do you tell whether a linear transformation is injective? Check its kernel: the transformation is injective exactly when the only vector it sends to zero is the zero vector itself.

Chapter 8: Representations

Difficulty: Hard · Key topics: vector representation, matrix representation, change-of-basis matrix, isomorphism of finite-dimensional spaces, normal matrices, orthonormal diagonalization

The final chapter makes precise an idea the book has been building toward since Chapter 2: once a vector space has a basis, every vector in it can be replaced by a column of coordinates, and every linear transformation between finite-dimensional spaces can be replaced by a matrix. Vector representation is shown to be itself a linear transformation, and in fact an isomorphism — which is why every n-dimensional vector space over the complex numbers behaves exactly like ordinary complex n-space, once a basis is fixed. The chapter’s Fundamental Theorem of Matrix Representation ties composing linear transformations to matrix multiplication, and a change-of-basis matrix is developed to translate a representation from one basis to another. The book closes with orthonormal diagonalization, applying everything — eigenvalues, bases, inner products — to the best-behaved case: normal matrices.

Key Points:

• Vector representation converts each vector into a column of coordinates relative to a chosen basis, and this conversion is itself an invertible linear transformation.
• Every n-dimensional vector space over the complex numbers is isomorphic to ordinary complex n-space, once a basis is fixed.
• The Fundamental Theorem of Matrix Representation connects composing linear transformations to multiplying their matrix representations.
• A change-of-basis matrix translates a vector’s coordinates from one basis to another without changing the vector itself.
• A matrix is normal when it commutes with its own adjoint; normal matrices are exactly the ones that admit an orthonormal basis of eigenvectors.
• Orthonormal diagonalization is the strongest form of diagonalization in the book, available for normal matrices — including every Hermitian and unitary matrix.

Memory Tip: Think of a basis as a translator — vector and matrix representation are both “write it down in this basis’s language,” and the change-of-basis matrix is simply the dictionary between two such languages.

Common Mistake: Treating a vector’s coordinate representation relative to a basis as if it were the vector itself, and forgetting the same vector looks different in a different basis.

Important Questions:

• What does the Fundamental Theorem of Matrix Representation connect? It connects the composition of two linear transformations to the matrix multiplication of their matrix representations, once bases are fixed for every space involved.
• What property makes a matrix admit an orthonormal basis of eigenvectors? The matrix must be normal — it commutes with its own adjoint — which includes every Hermitian and unitary matrix as a special case.

Appendix: Preliminaries

Appendix · Difficulty: Easy · Key topics: complex number arithmetic, complex conjugate, modulus, sets, subsets, set equality, cardinality, union and intersection

This appendix is reference material the book expects you to already know, collected here rather than in Chapter 1 because it is used unevenly throughout the book — some chapters need it heavily, others not at all. Complex number arithmetic is reviewed completely: addition, multiplication, the conjugate and the modulus, along with the basic algebraic properties the book proves once here and cites everywhere from Chapter 2’s inner products through Chapter 6’s eigenvalues. Set notation and vocabulary — subsets, equality, cardinality, union and intersection — get the same treatment, since the book states definitions and theorems with quantifiers and set language from the very first chapter.

Key Points:

• The complex conjugate of a+bi is a−bi; conjugating twice returns the original number, and conjugation respects both addition and multiplication.
• The modulus of a complex number is its distance from the origin in the complex plane, used throughout the book to build norms of complex vectors.
• A zero product of complex numbers forces at least one factor to be zero, exactly as with real numbers.
• Set equality means two sets contain exactly the same elements, regardless of how each set is described.
• Cardinality is the number of elements in a set, used later to compare the sizes of bases and vector spaces.
• Union and intersection combine sets the usual way, and both are used freely once formal proofs begin in Chapter 1.

Memory Tip: If a proof from Chapter 2 onward stalls on notation rather than ideas, it is worth a two-minute stop here — most notation snags trace back to a complex-number or set-theory convention defined on this page.

Common Mistake: Forgetting to conjugate when computing the modulus of a complex number — the modulus of a+bi is the square root of a²+b², and skipping the conjugate quietly breaks every norm and inner product computation downstream.

Important Questions:

• How do you compute the modulus of a complex number a+bi? As the square root of a²+b² — equivalently, the square root of the number times its own complex conjugate.
• What does it mean for two sets to be equal? That they contain exactly the same elements, no matter how each set is described or defined; order and repetition do not matter.

Download A First Course in Linear Algebra PDF (Free)

This book is free from its official source. Click below to open the official download page for A First Course in Linear Algebra, Version 3.50, where you can get the tablet, phone or print PDF, read the full online HTML edition, or download the separate exercise/solution manual.

How to Study This Book

The book is meant to be read in order — each chapter builds directly on the last — but not every course needs all eight chapters at the same depth. A typical one-semester Linear Algebra course in a BS Math or BSCS programme in Pakistan usually covers Chapters 1 through 6 in full and treats Chapters 7 and 8 more lightly, or leaves them for a second course. Read the chapter introductions first; each one states in a paragraph exactly why the chapter exists and how it connects to what came before, which makes the more abstract chapters far less intimidating.

Chapters 1 to 3 (Systems of Linear Equations, Vectors, Matrices) are the computational core. This is where you build fluency with row reduction, vector operations and matrix algebra, and it is worth over-practising here: every later chapter assumes these operations are automatic, not something you have to think through step by step.

Chapter 4 (Vector Spaces) is the hardest transition in the book — the point where “vector” stops meaning a column of numbers and starts meaning anything satisfying ten axioms. If proof-writing itself is new to you, read the book’s own Proof Techniques reference material (listed under Additional Resources on the official site) alongside this chapter rather than after it. Do not rush the subspace test; it is used constantly from here on.

Chapters 5 and 6 (Determinants, Eigenvalues) return to computation, but Chapter 6 is where complex numbers stop being optional — a real matrix can easily have complex eigenvalues. If your complex arithmetic is rusty, read the Preliminaries appendix before you start Chapter 6, not during it.

Chapters 7 and 8 (Linear Transformations, Representations) are the theoretical capstone and the most abstract material in the book. They reward a second pass: read once for the definitions, work several Archetypes by hand, then read again for the proofs. Many BS Math/BSCS programmes treat these two chapters as the bridge into a second, more theoretical linear algebra or abstract algebra course.

Use the 24 Archetypes deliberately. Every time a new theorem or technique appears, find the Archetype the section references and work it by hand before checking the book’s own worked version — this is the single most effective way to study this particular book, since the same handful of examples recur from Chapter 1 all the way to Chapter 8.

Do the exercises, and use their coding to pace yourself: exercises marked C are mostly computational, T are proof-based, and M mix the two. Work the three Reading Questions at the start of a section before reading the section itself — they are designed to be answerable only after you have understood the material, so attempting them first shows you exactly what to watch for as you read.

Used In These Programs

This book is used for the Linear Algebra course in: BS Mathematics · BS Computer Science. Browse all Linear Algebra books or all Mathematics books.

Who Should Read This

This book suits a BS Math or BSCS student taking their first university Linear Algebra course, whether the course is taught computationally or as a proof-based bridge course — the book genuinely supports both, since every computational technique is backed by a complete proof a student can read when they are ready for it. It assumes nothing beyond high-school algebra going in, reviews complex numbers and set notation in its own Preliminaries appendix, and is a strong choice for a student who has never written a mathematical proof before, since the definitions and proof techniques are introduced gradually rather than assumed. It is less suited to a student who wants only worked numerical examples with no justification — the proofs are the point of this book, not an optional extra — and a student in that position may prefer to lean on the 24 Archetypes as worked computational reference while treating the proofs as optional first-pass reading. Students preparing for graduate study in mathematics, or for a computer science specialisation that leans on linear algebra (graphics, machine learning, numerical methods), get particular value from the later chapters on linear transformations and representations, which most computationally-focused courses skip.

Applicable Universities

This book is useful for students at Pakistani universities offering BS Math and BSCS including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

FAQs

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A First Course in Linear Algebra is a rare thing on a free-textbook shelf: a fully proof-based, rigorous treatment that still works as a first course, backed by 24 worked Archetypes and 500+ exercises with a separate solution manual. It suits both a computational Linear Algebra course and a proof-based bridge course without needing a different book for either. Browse more Mathematics books for the rest of your semester.

A First Course in Linear Algebra, Version 3.50, by Robert A. Beezer, University of Puget Sound / Congruent Press. Free under the GNU Free Documentation License (GFDL), Version 1.2 or any later version, with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Official source: https://linear.pugetsound.edu/