Math 130 Linear Algebra
Please bookmark this page, http://aleph0.clarku.edu/~ma130/, so you can readily access it.
We won’t cover all of the topics listed below at the same depth. Some topics are fundamental and we’ll cover them in detail; others indicate further directions of study in linear algebra and we’ll treat them as surveys. Besides those topics listed below, we will discuss some applications of linear algebra to other parts of mathematics and statistics and to physical and social sciences.
Matrices and vectors
Matrix addition and scalar multiplication.
Matrix multiplication. Matrix algebra. Matrix inverses.
Powers of a matrix. The transpose and symmetric matrices.
Vectors: their addition, subtraction, and multiplication by scalars (i.e. real numbers). Graphical interpretation of these vector operations
Developing geometric insight.
Inner products and norms in Rn: inner products of vectors (also called dot products), norm of a vector (also called length), unit vectors.
Applications of inner products in Rn: lines,
planes in R3, and
lines and hyperplanes in Rn.
Systems of linear equations
Matrix inversion and determinants
Rank, range and linear equations
The rank of a matrix.
Rank and systems of linear equations.
Linear independence, bases and dimension
Basis and dimension in Rn.
Linear transformations and change of basis
Range and null space.
Change of basis and similarity
Eigenvalues and eigenvectors.
Diagonalisation of a square matrix.
When is diagonalisation possible?
Applications of diagonalisation
Powers of matrices.
Systems of difference equations.
Linear systems of differential equations.
Inner products and orthogonality
Gram-Schmidt orthonormalisation process
Orthogonal diagonalisation and its applications
Orthogonal diagonalisation of symmetric matrices.
Direct sums and projections
The direct sum of two subspaces.
Characterizing projections and orthogonal projections.
Orthogonal projection onto the range of a matrix. Minimizing the distance to a subspace.
Fitting functions to data: least squares approximation.
Complex matrices and vector spaces
Dave’s Short Course on Complex Numbers.
Complex vector spaces.
Complex inner product spaces.
Unitary diagonalisation and normal matrices.
Class notes, quizzes, tests, homework assignments
To be filled in as the course progresses.
Some old linear algebra tests
- First assignment. Exercises 1.1 through 1.7 page 53, and problems 1.1 through 1.6 page 55.
- Second assignment. Problems 1.8 through 1.14 page 57.
- Third assignment. Problems 2.1 through 2.8 page 86.
- Writing Proofs: structures of theorems and proofs,
synthetic and analytic proofs, logical symbols, and well-written proofs
- A little bit about sets
- Notes on Matlab
- Vector Spaces, their axiomatic definition
- Properties of vector spaces that follow from the axioms
- Subspaces of vector spaces
- Linear differential equations, an application of
- Linear combinations and basis
- Span, and independence
- The replacement theorem and dimension
- Where dimension doesn’t work
- Linear transformations, matrices, and linear operators
- Some linear transformations of the plane
- Composition and categories, composition of linear transformations and multiplication of matrices
- Algebra of linear transformations and matrices
- Kernel, image, nullity, and rank
- Change of coordinates
- Rank and nullity of matrices
- Introduction to determinants, 2x2 and 3x3 determinants, areas of triangles and parallelagrams in the plane, volumes of parallelepipeds, Jacobians
- Permutations and determinants
- Characterizing properties and constructions of determinants, cofactors, diagonal and triangular matrices
- More properties of determinants, an algorithm for evaluating determinants, determinants of products, inverses, and transposes, Cramer’s rule
- Eigenvalues, eigenvectors, and eigenspaces of linear operators
- Rotations and complex eigenvalues
- Diagonalizable operators and matrices
- Norm and inner products in Cn
and abstract inner product spaces
- Cauchy’s inequality
- Cross products
Web pages for related courses
Pages on the web that you may find interesting
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