| Topics | Linear algebra is a topic that is foundational to many areas of both pure and applied mathematics. The two main goals of this course are for you to reinforce and extend the concepts that you learned in Multivariable Mathematics I and improve your mathematical communication skills, in particular your proof-making and presentation skills. A rough outline of the course is:
- Vector space definitions and properties
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Linear map (aka linear transformation) definitions and properties
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Eigenvalues and eigenvectors
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Inner product space definitions and properties
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Operators on spaces
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Trace and determinant
You may notice that most of the stuff on that list is familiar from your work in previous courses. There are two main differences between this course and your earlier courses. First, the material is presented at a more abstract level rather than focusing primarily on Rn.
This will require you to pay careful attention to the definitions, theorems, and proofs that we will be discussing. Second, we will refrain from using the determinant until the end of the course. This will allow us to construct the topic in a way that more clearly illustrates the structure of linear algebra and sets the stage for the study of infinite dimensional vector spaces and functional analysis.
The learning objectives for the course are
- Gain a solid understanding of abstract vector spaces and linear maps.
- Understand the connection between eigenvalues, eigenvectors, and diagonal matrices.
- Gain an understanding of inner product spaces and operations on inner product spaces.
- Gain exposure to operators in complex vector spaces and real vector spaces.
- Be introduced to the concepts of trace and determinants of operators and matrices.
- Be able to prove results related to all of the above and communicate them both orally and written.
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