Linear Algebra Friedberg Solutions Chapter 3

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
  • Linear Algebra Friedberg Solutions Chapter 3

    Source: https://www1.cmc.edu/pages/faculty/bhunter/math115a/

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