Linear Algebra

About this book

Linear Algebra by David Cherney, Tom Denton, Rohit Thomas, and Andrew Waldron connects systems of equations with vector spaces and linear transformations. The linked first-edition UC Davis text was edited by Katrina Glaeser and Travis Scrimshaw. Its seventeen teaching chapters include optimization, geometry, eigenvectors, orthogonality, and approximation. This guide restores the complete author credit and follows the source’s chapter sequence, with separate attention to its review and assessment resources.

Chapter and section guide

1. What Is Linear Algebra?

Vectors and linear functions provide the organizing ideas. Matrices are not merely arrays to manipulate: they can encode information and represent transformations. The chapter connects several viewpoints so students can move between a system, a geometric picture, and an operation on vectors. Linearity means respecting addition and scalar multiplication, not simply producing a straight-looking graph. Students should test the definition directly before using matrix methods that assume it.

Study application: Give one linear and one nonlinear map with the same input and output spaces. Test addition and scalar multiplication explicitly and identify the failed condition.

2. Systems of Linear Equations

Gaussian elimination transforms a system into an equivalent form that reveals consistency and freedom. Row operations preserve the solution set, while pivots and free variables organize its description. A system may have no solution, one solution, or a family of solutions. Students should distinguish the coefficient matrix from the augmented matrix and record the operations used. A numerical answer should be checked in the original equations.

Study application: Row-reduce a small system with a free variable and write its complete solution set. Explain how the pivot structure reveals the freedom rather than choosing one arbitrary solution.

3. The Simplex Method

Linear programming replaces equality-only questions with an objective subject to constraints. The feasible region and the quantity to optimize must be stated separately. The chapter introduces a structured method for moving among candidate solutions. Students should understand the geometric reason for examining boundary configurations and recognize that infeasibility and unboundedness differ from an ordinary optimum. A computed candidate must satisfy the original constraints.

Study application: Sketch a two-variable feasible region and evaluate a linear objective at relevant corners. Explain how a change in a constraint could make the problem infeasible or unbounded.

4. Vectors in Space and n-Vectors

Vector arithmetic extends geometric reasoning into spaces with more coordinates. Components depend on a chosen representation, while addition and scaling describe structured combinations. Dot products and geometry connect algebra with lengths and angles in familiar settings. Students should identify the dimension and coordinate convention before combining objects. A list of numbers can represent many physical or abstract quantities, so interpretation comes from the defined space and operations.

Study application: Express a geometric vector using coordinates and verify a length or angle calculation. Then explain what changes when an n-vector represents data rather than a physical displacement.

5. Vector Spaces

A vector space is defined by its operations and axioms rather than by visual resemblance to ordinary arrows. Examples can include functions, polynomials, and other mathematical objects. Students should test closure and the required algebraic properties before applying familiar vector-space conclusions. A subset can fail to be a space even if many of its elements look familiar. The chapter broadens the meaning of linear algebra without abandoning precise definitions.

Study application: Test whether a specified collection of polynomials forms a subspace under the usual operations. Identify a decisive failed condition when the answer is negative.

6. Linear Transformations

A linear transformation respects vector-space structure. Its action on a spanning collection can determine its action more broadly, which prepares the matrix viewpoint. Students should distinguish a transformation from the matrix representing it in chosen bases. Composition and properties such as injectivity should be interpreted through what happens to vectors. A formula with a nonzero constant offset is generally affine rather than linear.

Study application: Define a transformation on basis vectors and compute its action on another vector. Explain why the basis representation makes the answer determined.

7. Matrices

Matrix arithmetic organizes transformations, systems, and composition. Multiplication is constrained by dimensions and is generally noncommutative. Invertibility means a transformation can be reversed on the relevant spaces; it is not a property shared by every matrix. The chapter connects calculations with their structural meaning. Students should predict the size of a result and explain an entry before relying on a mechanical algorithm.

Study application: Compute two composable transformations in both orders when dimensions allow. Explain geometrically or algebraically why the results can differ.

8. Determinants

The determinant links algebraic tests with geometric scaling and orientation. For a square matrix, a zero determinant signals loss of invertibility. Determinants behave predictably under row operations, but careless use of those operations can change the value. Students should distinguish the determinant from the matrix and understand why it is not defined as an ordinary determinant for every rectangular array. Geometry can check the plausibility of a result.

Study application: Compare a transformation that stretches area with one that collapses the plane onto a line. Explain their determinants and the relationship with invertibility.

9. Subspaces and Spanning Sets

A span consists of all allowed linear combinations of a collection. Subspaces capture the structure of solution sets and transformation outputs. Students should distinguish an individual vector, a set of generators, and the entire space they generate. Adding a vector already in the span does not enlarge it. The chapter prepares basis and dimension by making redundancy and reach explicit.

Study application: Give two different spanning sets for the same subspace. Explain why one can contain more vectors without describing a larger space.

10. Linear Independence

Independence asks whether a linear combination equals zero only through the trivial coefficients. It is a statement about a collection, not a judgment that one vector looks different from another. Dependence reveals redundancy and can be expressed by writing one vector in terms of others. Students should connect the algebraic test with pivots and the number of independent directions available. Pairwise difference does not establish independence of the whole set.

Study application: Construct three distinct nonzero vectors in a plane and demonstrate their dependence. Explain why no pair being scalar multiples would still be insufficient to make all three independent.

11. Basis and Dimension

A basis combines spanning with independence, giving each vector a unique coordinate description. Dimension counts basis elements, not necessarily the number of entries in one particular representation. Changing a basis changes coordinates while preserving the underlying vector. Students should justify both parts of a basis claim rather than rely on counting alone. This chapter turns redundancy analysis into an organized representation system.

Study application: Find a basis for a small subspace, express a vector in that basis, and verify uniqueness through independence.

12. Eigenvalues and Eigenvectors

An eigenvector retains its direction under a transformation, up to multiplication by an eigenvalue. The zero vector is excluded from the definition. The characteristic equation provides candidates for square matrices, but each candidate requires an associated eigenspace calculation. Students should distinguish an eigenvalue from a vector and interpret negative or zero values through the transformation. Not every matrix has a real eigenbasis.

Study application: Compute a simple matrix’s eigenvectors and verify Av=λv directly. Explain what a zero eigenvalue says about the transformation’s kernel.

13. Diagonalization

Diagonalization represents a transformation in a basis of eigenvectors when enough independent eigenvectors exist. It can simplify powers and reveal separate modes of action. Having eigenvalues does not automatically guarantee diagonalizability, and repeated eigenvalues require examination of eigenspace dimensions. Students should connect the change-of-basis matrix with the transformation rather than memorize a factorization alone.

Study application: Compare a diagonalizable matrix and a small defective example. Explain the missing independent direction in the latter and why writing down repeated eigenvalues does not solve the problem.

14. Orthonormal Bases and Complements

Orthogonality connects geometry with efficient coordinates and projections. An orthonormal basis makes coefficient calculations especially simple, while orthogonal complements describe directions perpendicular to a subspace. Students should distinguish orthogonal from normalized and track the inner product being used. The chapter’s constructions require appropriate hypotheses and cannot divide by a zero vector. A projection should be verified by checking the residual’s orthogonality.

Study application: Project a vector onto a line, then compute the residual. Check that the residual is perpendicular to the line and explain the resulting decomposition.

15. Symmetric Matrices

Real symmetric matrices have special spectral properties that support an orthonormal eigenbasis. The chapter connects this structure with a particularly clear geometric decomposition. Students should identify symmetry before applying the conclusion; a general square matrix need not share it. Orthogonal diagonalization preserves geometric interpretation while separating the transformation into principal directions. A calculation should verify both eigenvector relations and orthonormality.

Study application: Diagonalize a simple real symmetric matrix using normalized eigenvectors. Explain why the change-of-basis matrix is orthogonal and how that differs from an arbitrary diagonalization.

16. Kernel, Range, Nullity, and Rank

The kernel records inputs mapped to zero, while the range records attainable outputs. Rank and nullity connect these spaces with the domain’s dimension. Students should identify which space each object belongs to and avoid treating row and column information as interchangeable without explanation. The rank–nullity relationship organizes solvability and uniqueness. A nontrivial kernel means distinct inputs can have the same output.

Study application: For a rectangular matrix, find bases for the kernel and range and verify rank plus nullity equals the domain dimension.

17. Least Squares and Singular Values

An inconsistent system can still support an approximation problem. Least squares seeks a residual of minimal length under the stated norm, and singular-value structure provides another view of a transformation’s action. Students should distinguish an exact solution from a best approximation and examine uniqueness conditions. A small residual alone does not validate the modeling assumptions or make a prediction reliable outside the data. Numerical sensitivity remains part of interpretation.

Study application: Fit a line to a small original dataset and inspect residuals. Explain the optimization being solved and why minimizing squared error is different from proving the line is a correct causal model.

Supplementary Material and Assessments

Fields, symbols, online resources, sample examinations, and movie scripts extend the chapters. These resources serve different purposes: an assessment checks integrated skill, while a script or illustration may clarify one idea. Students should attempt review work before reading solutions and identify the concept behind each calculation. The scripts should not be assumed to have been executed merely because their text is available.

Study application: Use a sample assessment as a diagnostic and classify errors as definitional, conceptual, arithmetic, or representational. Rework one error from each category with an explanation.

Study instructions

1. For each object, state its space, dimensions, and the operations being used.
2. Connect row reduction, transformations, and geometry instead of treating them as unrelated chapters.
3. Prove a small claim from definitions before relying on a memorized test.
4. Verify calculations by substitution, dimensions, invariants, and geometric interpretation.
5. Use sample examinations to identify conceptual gaps, then revisit the relevant chapter before repeating similar calculations.

Practice and review task

Analyze one small matrix from several viewpoints: solve a system, find its kernel and range, test invertibility, examine eigenstructure, and interpret its geometric action. Explain which conclusions depend on the matrix being square or symmetric.

Questions for reflection

Why can a transformation remain the same when its matrix changes with a basis? What separates a best-fit solution from an exact solution and from a scientifically valid model?

Sources and coverage

This original study guide follows First edition, UC Davis, 2013. It draws on the source contents, chapter introductions, selected explanations and examples, and review material. It condenses the resource rather than replacing the full text; study applications are original exercises. The source title page credits all four authors; Katrina Glaeser and Travis Scrimshaw are credited as editors. The imported author omission and Waldon misspelling are corrected.

Reviewed source text; Book and cover catalog record.