Sections, concepts, and problems

6.1. Inner products and norms. Inner products and their properties, two standard examples (the standard inner product on Fⁿ and the Frobenius inner product on the set of n by n matrices over F), the conjugate transpose of a matrix, the norm of a vector in an inner product space, orthogonal vectors, unit vectors, orthonormal vectors.
1, 3, 8 (use Thm 6.1 at some point), 10, 11, 15a (just do what they say), 17 (recall Thm 2.4), 22a

6.2. Gram-Schmidt orthogonalization process. Orthonormal bases, why we care about them, how to transform an arbitrary basis into an orthogonal basis, Fourier coefficients, orthogonal complements.
1, 2ad, 4, 5, 6, 7, 9, 13abc,18

6.3. The adjoint of a linear operator. (Assume V is finite-dimensional.) How every linear functional is an inner product with a fixed y in V, what an adjoint of a linear operator on V is, what it has to do with conjugate transposes of matrices, properties of adjoints, and an application to least squares approximations.
1, 2 (note: for (c), use the orthonormal basis given in Section 6.2 Problem 2(c) to save yourself a ton of work), 3, 4, 6, 7 (try the linear operator on R² that reflects about the line y=x and then projects onto the x-axis), 12a, 20a(i)

6.4. Normal and self-adjoint operators. Normal linear operators and matrices, various properties of these, what normal operators have to do with orthonormal bases and eigenvectors, self-adjoint (or Hermitian [her-me-shun]) linear operators and matrices, and various properties of these.
1, 2ac, 4, 5, 9 (part a of 6.15 is what matters), 12 (see Thm 6.16 ... it starts by noticing that the characteristic poly splits), 17a, 20

6.5. Unitary and orthogonal operators and their matrices. Unitary/orthogonal operators, properties they possess, conditions their eigenvalues have, orthogonal/unitary matrices, examples of rotations and reflections of the plane, unitary/orthogonal equivalence of matrices, rigid motions, and an application to quadratic forms.
1, 2b, 7 (hint: Cor. 2; remember i²=-1), 9 (try U(z,w)=(z+w,0) on C² with respect to the standard basis), 18 (piece of cake), 22, 24, 27a

Other Chapters

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