Readings

Most of the reading assignments below refer to the textbook:

[Axler] = Axler, Sheldon J. Linear Algebra Done Right. Springer, 2004. ISBN: 9780387982588. [Preview with Google Books]

A few topics not fully covered in the textbook are discussed in supplementary notes available in the Study Materials section. These include:

[F] = Notes on Finite Fields

[OSI] = Notes on One-sided Inverses

[GE] = Notes on Gaussian Elimination

[OB] = Notes on Orthogonal Bases

[ST] = Proof of the Spectral Theorem

[EIG] = Notes on Generalized Eigenvalues

LEC # TOPICS READINGS
1 Definition of vector spaces, properties [Axler] Chapter 1, pp. 2–12.
2 Subspaces, sums and direct sums [Axler] Chapter 1, pp. 13–18.
3 Span, independence, bases [Axler] Chapter 2, pp. 21–31.
4 Bases and dimension [Axler] Chapter 2, pp. 31–34.
5 Linear maps, null space / range [Axler] Chapter 3, pp. 38–47.
6 Matrices, invertibility [Axler] Chapter 3, pp. 48–58.
7 Exam 1 on Chapters 1–3 No Readings
8 Finite fields. Systems of equations. [F], [GE], [OSI]
9 Gaussian elimination [GE]
10 Counting over Fp. Invariant subspaces. [Axler] Chapter 5, pp. 75–79, [F]
11 Finding eigenvectors [Axler] Chapter 5, pp. 80–81.
12 Upper triangular and diagonal matrices [Axler] Chapter 5, pp. 81–90.
13 Eigenvectors over R and Fp [Axler] Chapter 5, pp. 91–93, [F]
14 Exam 2 on Chapters 1–5 No Readings
15 Inner products, Gram-Schmidt [Axler] Chapter 6, pp. 97–111, [OB]
16 Orthogonal projection, minimization [Axler] Chapter 6, pp. 111–16.
17 Adjoint, self-adjoint, normal [Axler] Chapter 7, pp. 127–32.
18 Spectral theorem [Axler] Chapter 7, pp. 132–44, [ST]
19 Positive operators [Axler] Chapter 7, pp. 144–47.
20 Isometries, polar decomposition [Axler] Chapter 7, pp. 147–57.
21 Exam 3 on Chapters 1–7 No Readings
22 Generalized eigenspaces [Axler] Chapter 8, pp. 163–68, [EIG]
23 Generalized eigenspace decomposition [Axler] Chapter 8, pp. 173–78, [EIG]
24 Characteristic polynomial [Axler] Chapter 8, pp. 168–73.
25 Determinant [Axler] Chapter 10.
26 Trace, canonical commutation relations [Axler] Chapter 10.