Homework Homework will be assigned each Monday, Tuesday, Wednesday, and Thursday and will be due at the beginning of class each Monday. Weekly Reading Week #1 - Monday, August 6th
Sup Ch 2
Ch 1.1 -1.5
Sup Ch 3
Week #2 - Monday, August 13th
Ch 1.5 and 1.6
Ch 2.1 - 2.3
Week #3 - Monday, August 20th
Ch 2.1-2.4
Appendix D (Fundamentak Theorem of Algebra)
Notes on Methods of Proofs by a former 115A student can be found here.
Week #4 - Monday, August 27th
Ch 4.1 and 4.4
Ch 5.1 and 5.2
Appendix D (Complex Numbers and The Fundamentak Theorem of Algebra) Week #5 - Tuesday, September 4th NO CLASS Monday (Labor Day)
Ch 4.4 - Important Facts about Determinants
Ch 5.2 - Diagonalizability
Ch 6.1 and 6.2 - Inner Products and Norms Week #6 - Monday, September 10th FINAL EXAM Thursday September 13th
Ch 6.2 - Inner Products, Norms, The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
Ch 6.3 - The Adjoint of a Linear Operator
Ch 6.4 - Inner Products and Norms
Ch 6.6 - The Spectral Theorem
Ch 6.7 - Singular Value Decompostion (SVD)
Review Practice Midterm A short practice midterm can be found here.
The solutions to the practice midterm can be found here.
An old midterm with solutions can be found here (this was a 1 hour exam covering similar material, your's will be longer).
Solutions to the midterm.. Practice Final A short practice final can be found here.
The solutions to the practice final can be found here.
An old final with solutions can be found here (this was a 1 hour exam covering similar material, your's will be longer).
Solutions to the final.. Homework Assignments HW #1 - due Monday, August 13th
Sup Ch 2.1, 2.4, 2.5a, 2.6a, 2.8, 2.9, 2.10
Ch 1.2 Exercies 8, 13, 17,
Ch 1.3 Exercies 4, 19, 20 (Hint: read Sup Chapter 3 on induction first), 23 Ch 1.4 Exercies 12,
Ch 1.5 Exercies 2a,c,e, 12, 17
Homework assigned in lecture can be found here.
Solution can be found here.
HW #2 - due Monday, August 20th
Ch 1.6 Exercies 2a,b,c, 7, 13, 19 and What are the dimensions of W1 and W2 of exercise 14?
Ch 2.1 Exercies 1, 2, 8, 15, 18, 19, 21a, 25a
Let T: V --> W be a linear map, show the null space, null(T), is a subspace of V.
Ch 2.2 Exercies 1, 4, 5a, 16
Solution can be found here.
HW #3 - due Monday, August 27th
Ch 2.2 Exercies 10, 12
Ch 2.3 Exercies 1, 2a, 10, 12, 13, 14, 16, 20, 23
Ch 2.4 Exercies 1, 3, 4, 6, 7, 9, 15, 17
Two proofs from class,
1 - Let U, V, W be vector spaces over a field F, and suppose that the linear maps S:U-->V and T:V-->W are both one-to-one. Prove the composition ToS is one-to-one (injective).
2 - Let V be a finite dim. vector space over a field F, with linear maps S:V-->V and T:V-->V. Prove the composition ToS is invertible if and only if both S and T are invertible.
Solution can be found here.
HW #4 - due Tuesday, September 4th
Ch 4.1 Exercies 2a, 3b, 4c, 7, 9
Ch 4.2 Exercies 5, 9, 14
Ch 4.3 Exercies 9, 11, 13b
Ch 5.1 Exercies 3b, 3c, 9, 14, 18b
Solution can be found here. HW #5 - due Monday, September 10th Ch 5.2 Exercies 2c,f, 11b, 12a, 12b
Prove: If A and B are similar nxn matrices then A and B have the same eigenvalues.
Ch 6.1 Exercies 2, 9, 20, 24a,d, 25
Prove the Parallelogram Law: A normed vector space X, is an inner product space with a norm defined from the inner product ||x||=sqrt() if and only if
||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2 for all x,y in X. HW #6 - due Thursday, September 13th Ch 6.2 Exercies 2b,h, 4
Ch 6.4 Exercies 2a,f, 6a,c (For both 2 and 6 determine if T is normal, self-adjoint or neither, you don't need to find an eigen basis.)
Ch 6.5 Exercies 17
Ch 6.7 Exercies 3b,e, 6b,e, 18a
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