**APSC 174**

**Introduction To Linear Algebra**

**Units: 3.30**

Systems of linear equations; real vectors spaces and subspaces; linear combinations and linear spans; linear dependence and linear independence; applications to systems of linear equations and their solution via Gaussian elimination; bases and dimension of real vector spaces; linear transformations, range, kernel and Rank-Nullity theorem; matrix representation of a linear transformation; composition of linear transformations and matrix multiplication; invertible matrices and determinants; eigenvalues and eigenvectors of square matrices. Applications of the course material to engineering systems are illustrated.

(Lec: 2.8, Lab: 0, Tut: 0.5)

(Lec: 2.8, Lab: 0, Tut: 0.5)

**Offering Term:**WS

**CEAB Units:**

Mathematics 40

Natural Sciences 0

Complementary Studies 0

Engineering Science 0

Engineering Design 0

**Course Equivalencies:**MATH 110B/112 / APSC 174

**Offering Faculty:**Smith Engineering

**Course Learning Outcomes:**

- Solve parametrized and unparametrized systems of linear equations using Gaussian elimination and back substitution by applying elementary row operations on an augmented matrix; for parametrized systems also determine the number of solutions as a function of the parameter.
- Perform basic matrix algebraic operations (addition, scaling, multiplication), and compute and utilize properties of the determinant of a real n x n matrix (including using it to assess whether a matrix is invertible).
- Explain the mathematical concept of a real vector space, and determine whether a given subset of a vector space is a vector subspace by working with both the usual Euclidean space Rn and other vector spaces.
- Demonstrate an understanding of linear combination, linear dependence, linear span, basis and dimension by: i) determining whether a given vector in is the linear span of a family of vectors and whether that family of vectors is linearly independent, ii) computing a basis for a given vector space and its dimension.
- Define a linear mapping between vector spaces and determine if a given mapping is linear.
- Define the kernel and image of a linear mapping, compute them for a given real matrix, and explain how they are related to the column vectors of that matrix.
- Define eigenvalues, eigenspaces and eigenvectors for a given vector space endomorphism and compute them for a real n x n matrix.
- Prove linear algebraic results for general vector spaces with mathematical reasoning and in precise mathematical language, combining concepts such as vector subspaces, linear span, linear independence, linear mapping, and eigenvalues/eigenspaces/eigenvectors.