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# Applied Linear Algebra 2

This is the courseware for MATH 225: Applied Linear Algebra 2 at the University of Waterloo.

Topics include a continued discussion on vector spaces. Linear transformations and matrices are examined more fully. Inner products, eigenvalues and eigenvectors, and diagonalization are introduced and disucssed. Applications to linear algebra are also examined.

### Units

## Vector Spaces

In this module, we take the basic properties of R^{n} and create an object called a vector space. We then look at how the familiar concepts of subspace, span, linear independence, and basis behave in general vector spaces. Examples from R^{n}, matrices, and the newly introduced polynomial spaces are used throughout these discussions.

## Orthonormal Bases

In this module, we explore some properties of the “standard basis” for our known vector spaces, and generalize this to the idea of an orthonormal basis. We will learn the Gram-Schmidt procedure for finding an orthonormal basis, and will end by generalizing the idea of an orthonormal basis even further by generalizing the idea of the dot product in R^{n} to the concept of an inner product.

## Complex Vector Spaces

In this module, we expand our study of vector spaces by changing the numbers used for scalars from the real numbers to the complex numbers. We start with an introduction to the complex numbers, and then begin to look at the vector space C^{n}. The topics of subspace, span, linear independence, and basis will be discussed in this setting, as well as orthonormal bases and inner product spaces. We also learn how to use complex numbers to diagonalize matrices from the real numbers.