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Math 122

Introduction to Linear Algebra

Calendar Description: Vector geometry in R2 and R3 , the vector space Rn and its subspaces, spanning
sets, linear independence, bases and dimension, dot product in Rn , systems of linear equations and
Gaussian elimination, matrices and matrix operations, matrix inverse, matrix rank, linear transformations
in Rn , introduction to determinants, Cramer’s rule, introduction to eigenvalues, eigenvectors and
diagonalization of real matrices, applications of linear algebra.
Relation to Other Courses: MA122 is the prerequisite for MA201 Multivariable Calculus, MA222
Linear Algebra, MA270 Financial Math I, MA305 Diﬀerential Equations II, MA307 Numerical Analysis,
MA343 Introduction to Multivariate Analysis, ST230 Introduction to Probability and Statistics for
Science, ST259 Probability I and ST362 Regression Analysis. Some of these courses in turn provide
prerequisites for higher level courses in mathematics and statistics.
Tentative	Schedule: (references are to the textbook listed above)
Lectures	Topic	Course notes
1-5	Euclidean n-space: lines, planes, areas, orthogonal projection, cross prod- uct	s1.1 — 1.6
6-7	Linear systems: Gauss elimination and Gauss-Jordan elimination	s2.1 — 2.2
8-11	Matrix algebra: multiplication, invertibility, elementary matrices	s3.1 — 3.5
12-14	Determinants: deﬁnition and properties, adjoint matrix, Cramer’s rule,	s4.1 — 4.4
	cross product
15-19	Subspaces and basis: subspaces and linear span, linear independence, basis	s5.1 — 5.4
	and dimension, coordinates
20-21	Matrix transformations, special linear operations in R2 and R3	s6.1 — 6.2
22-23	Eigenvalues and eigenvectors: deﬁnition, diagonalizability and diagonal- ization	s7.1 — 7.3
24	Review

Department of Mathematics, Faculty of Science, Wilfrid Laurier University (Waterloo campus)

Learning outcomes: Linear algebra is essentially the study of vectors, matrices and linear mappings. Since linear structure is a geometric model of natural and social phenomena using the simplest algebraic computations, linear algebra plays an extremely important role in almost every branch of natural and life sciences, social sciences, business and management as well as mathematics itself. This introductory course deals with only Euclidean real spaces. Abstract concepts are introduced with geometric intuition. General patterns of applications are summarized in the form of algorithms or recipes. General theorems are all illustrated by concrete examples, together with proofs in a few steps or by algorithms. The aim of the

course is to provide students with tools that can be applied right away in higher level mathematics courses

and in other disciplines, and to prepare students for a more abstract and systematic study of linear algebra

in the course MA222. At the end of the course, students are expected to be able to:

· Solve geometric problems concerning lines and planes in Euclidean spaces in terms of their algebraic equations in vector and scalar forms.

· Use the dot product and cross product of vectors to interpret and solve geometric problems.

· Determine whether a set of vectors is linearly independent and determine whether a subset of a Euclidean space is not a subspace.

· Find a basis for a subspace of a Euclidean space; in particular, for the solution space of a homogeneous system of linear equations.

· Determine whether a square matrix is invertible and, if it is, ﬁnd its inverse.

· Perform matrix operations and compute determinants of square matrices.

· Interpret and solve problems related to linear systems and matrix transformations using matrices and their properties.

· Find real eigenvalues of a square matrix and their corresponding eigenspaces.

· Determine whether a square matrix is diagonalizable.