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Linear Algebra 1
This is the courseware for MATH 136: Linear Algebra 1 for Honours Mathematics at the University of Waterloo.
Topics include systems of linear equations, matrix algebra, elementary matrices, and computational issues. Other areas of the course focus on the real n-space, vector spaces and subspaces, basis and dimension, rank of a matrix, linear transformations and matrix representations. Determinants, eigenvalues and diagonalization, and their applications are also explored.
Units
Vectors in Euclidean Space
In this module, we will look at basic properties of vectors in Euclidean space. This setting will allow us to use geometric interpretations to introduce the important concepts of spanning, linear independence, and bases.
Systems of Linear Equations
Already we have seen many cases in linear algebra where it is required to solve m equations in n unknowns — for example, when determining whether a vector is in the span of a set of vectors, when determining if a set of vectors is linearly independent, or when calculating the formula for the cross product. We will see that there are many other cases where we need to do this.
Additionally, such problems not only arise in linear algebra, but in many other areas of mathematics, science, economics, business, et cetera. In real-world situations, we could easily have thousands of equations and thousands of variables. Thus, it is important to learn and understand the theory behind this, and not just simply memorize the method for solving small systems.
Matrices and Linear Mappings
In this module, we will look at matrices as objects rather than just as a tool for solving systems of linear equations. In doing so, we will be led to the idea of linear mappings.
Vector Spaces
In this module, we will extend lots of what we did with vectors in Rn to general vector spaces. In particular, we will look at spanning, linear independence, subspaces, and bases. We will then use this theory to precisely define dimension, and to look at coordinates of a vector with respect to a basis.
Inverses and Determinants
In this module, we will look at properties of invertible matrices, how to determine if a matrix is invertible using the determinant, and some of the uses of invertible matrices and determinants.
Diagonalization
We now look at one of the most important and useful pieces of linear algebra — diagonalization. In this module, we will need to use material from every module.