Mathematics 535 Linear Algebra Homework

You are required to re-submit homework problems until you have solved them correctly if you are to get credit for them.

  1. Review Assignment Due on September 8, 2008.

Problems marked with a * are extra credit, and carry the weight of two ordinary problems. Problems are due the Monday after the section has been discussed in class unless noted otherwise.

  1. Linear Equations
    1. Fields: Page 5, Number 8.
    2. Systems of Linear Equations: Page 5, Numbers 2, 6.
    3. Matrices and Elementary Row Operations: Page 11, Numbers 3, 7, 8.
    4. Row-Reduced Echelon Matrices: Page 16, Numbers 3, 8.
    5. Matrix Multiplication: Page 21, Numbers 6, 7.
    6. Invertible Matrices: Page 26, Number 4. Page 27 Numbers 8, 12*.
  2. Vector Spaces
    1. Vector Spaces: Page 34, Numbers 6, 7.
    2. Subspaces: Page 40, Numbers 8, 9.
    3. Bases and Dimension: Page 48, Number 7. Page 49, Number 9.
    4. Coordinates: Paage 54, Number 1. Page 55, Number 6.
    5. Summary of Row Equivalence
    6. Computations Concerning Subspaces: Page 66, Numbers 2, 3, 4.
  3. Linear Transformations
    1. Linear Transformations: Page 73, Numbers 3, 6, 7. Page 74, Number 13.
    2. The Algebra of Linear Transformations: Page 84, Numbers 8, 9.
    3. Isomorphism: Page 85, Number 1. Page 86, Number 3.
    4. Representation of Transformations by Matrices: Page 95, Numbers 1, 4, 5, 6. Page 96, Numbers 8, 12.
    5. Linear Functionals: Page 105, Numbers 1, 3, 4. Page 106, Numbers 11, 12.
    6. The Double Dual: Page 111, Number 2*.
    7. The Transpose of a Linear Transformation: Page 115, Number 2. Page 116, Number 6.
  4. Polynomials
    1. Algebras
    2. The Algebra of Polynomials
    3. Lagrange Interpolation: Page 126, Number 1. Page 127, Number 4, 6*
    4. Polynomial Ideals: Page 134, Number 3, 5.
    5. The Prime Factorization of a Polynomial
  5. Determinants
    1. Commutative Rings
    2. Determinant Functions: Page 148, Number 3. Page 149, Numbers 5, 9, 11, 12.
    3. Permutations and Uniqueness of Determinants: Page 155, Numbers 3, 5, 7.
    4. Additional Properties of Determinants: Page 162, Numbers 1, 2a, 3, 4. Page 163, Numbers 5, 6, 10, 14*.
  6. Chapter 6: Elementary Canonical Forms I
    1. Introduction
    2. Characteristic Values (Eigenvalues): Page 189, Number 3. Page 190, Numbers 7, 8, 9, 13*.
    3. Annihilating Polynomials: Page 198, Numbers 2, 5, 9.
    4. Invariant Subspaces: Page 205, Numbers 2, 3, 5.
  7. Chapter 8: Inner Product Spaces
    1. Inner Products: Page 275 Numbers 1, 5, 6. Page 276, Numbers 9, 11.
    2. Inner Product Spaces: Page 289, Numbers 4, 5, 9. Page 290, Number 17.
    3. Linear Functionals and Adjoints. Page 298, Numbers 2, 4, 7. Page 299, Numbers 8, 9.
    4. Unitary Operators: Page 308, Number 3. Page 309, Number 7. Page 311, Number 15.
    5. Normal Operators: Page 317, Numbers 4, 8, 9, 10.
  8. Chapter 7: Elementary Canonical Forms II
    1. Simultaneous Triangulation and Diagonalization
    2. Direct Sum Decomposition
    3. Invariant Direct Sums
    4. The Primary Decomposition Theorem
    5. The Jordan Form: Page 250, Numbers 3, 5. Page 251, Numbers 15, 16.