Linear Algebra

These are the homework assignments completed in LaTeX for my linear algebra course MATH2101 Summer 2016. Each set consists of five major questions with numerous sub-questions. Some of the sets are more theoretical, and some are more applied. In most cases question 5 is a set of review exercises.

Assignment set instructions

Assignment 1 (source)

Vectors

1. Arithmatic operations on vectors.
2. Properties of dot products. proof of the dot product / cosine relationship.
3. Parameterization of lines and planes, distance between a line and a point.
4. Using elementary row operations to solve systems of equations.
5. Exercises: Parameterization of non-unique solutions to systems of equations.

Assignment 2 (source)

Matrices and applications

1. Solving systems of equations using row echelon and reduced row echelon form of matrices.
2. Deriving the normal vector, proof that the cross product is perpendicular.
3. Application: Stochastic modeling.
4. Application: Discrete time modeling.
5. Exercises: Finding angle between vectors, unit vectors, normal vector.

Assignment 3 (source)

Linear Transformations

In this workset I made extensive use of TikZ graphics to produce transformation diagrams.

1. Linear transformations. Proof that linear transformations are linear operators.
2. Proof that linear transformations are not commutative.
3. Application: Stochastic modeling with linear transformations.
4. Inverses of linear transformations. Proof of the invertibility of linear transformations.
5. Exercises: Row reduction, Proof of Cauchy-Schwartz inequality, Proof of commutativity of vector dot product.

Assignment 4 (source)

Vector Spaces

1. Exercises: Elementary Matrix operations.
2. Left and Right inverses, deriving the determinant.
3. Proofs of properties of Matrices.
4. Properties of vector spaces. Proofs that various sets are or are not vector spaces.
5. Exercises: Vector spaces.

Assignment 5 (source)

Basis, Span, Independence

This was the most difficult and proof-heavy assignment of the course.

1. Exercises: Find row-space, column-space, and null-space of matrices.
2. Proofs that the span of a vector space is linearly independent.
3. Proofs that independence and singularity of Ax=0 are correlated.
4. Gram-schmidt orthogonalization process. Proofs of orthogonality.
5. Exercises.

Assignment 6 (source)

Linear Mapping, Eigenvectors

1. Proofs whether various mappings are / are not linear.
2. Application: Permutation matrices.
3. Linear Transformations: Homogeneous coordinates.
4. Eigen vector / value pairs. Proofs of properties of Eigen vectors / values.
5. Exercises.

Assignment 7 (source)

Cramer's Rule

1. Exercises: Finding determinants.
2. Deriving Cramer's Rule.
3. Exercises: Finding eigen values / eigen vectors.
4. Application: Time series model approximation using eigen values / vectors.
5. Exercises.

Assignment 8 (source)

Lattices, The Wronskian

1. Exercises: Distances between points, lines, planes.
2. Application: Adjacency matrix of a graph.
3. Exercises: Finding basis and determining linear independence.
4. Lattices: Closest vector problem.
5. Exercises.