Sections, concepts, and problems

5.1. Eigenvalues and eigenvectors. Diagonalizable matrices, eigenvalues, eigenvectors, characteristic polynomials, and a geometric description of a few low-dimensional examples.
1, 2abd, 3, 5, 9, 11, 12, 15, 26 (see Ex. 4 pg. 553 if you don't know what Z₂ is)

5.2. Diagonalizability. The point of this section is to determine a relatively easy way to tell when a linear operator (or a square matrix) is diagonalizable; Thm 5.9 is the answer, and this is summarized on page 269. Along the way, the notions of when a polynomial splits over a field, multiplicity of roots of a polynomial, and the eigenspace of a linear operator (or a square matrix) are introduced. There is also a small subsection on applications to systems of DEs, like you once saw in MATH 238. You need not read the optional subsection on direct sums; we may come back to that later.
1a-g, 2acg, 3ad, 7 (see Ex. 7), 8, 9, 10, 11 (9-11 are all related)

5.4. Invariant subspaces and the Cayley-Hamilton Theorem. Spaces invariant under a linear operator T, T-cyclic subspaces generated by a single vector (compare with cyclic subgroups), what the characteristic polynomial of a T-invariant subspace has to do with the characteristic polynomial of T, bases and characteristic polynomials for T-cyclic subspaces, and the Cayley-Hamilton Theorem. You need not read the subsection on direct sums.
1a-f, 2, 3, 4, 5, 6, 11, 13, 14; (#23: 10 points extra credit)

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Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 | Chapter 5 | Chapter 6