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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.

 

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.

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.