Math 22

Linear Algebra with Applications


General Information Syllabus Homework


Homework


For the homework policy, please consult the General Information page.


For those students who have ordered the textbook online and not yet received it: (These are PDF files, requiring Adobe Reader)

Solutions to Assignment 3.
Solutions to Assignment 5.
Solutions to Assignment 6.
Solutions to Assignment 7.


Assignment 1

(Due Wednesday, June 29)
Section 1.1: Systems of Linear Equations
  • Reading Goals: Write a system of linear equations in matrix notation; solve a linear system using elementary row operations; determine whether a system is consistent.
  • Required Problems: see handout
Section 1.2: Row Reduction, Echelon Forms
  • Reading Goals: Compute the echelon and reduced echelon forms of a matrix and use them to solve systems of equations; determine whether a solution is unique, and if not, parameterize the set of all solutions.
  • Required Problems: see handout


Assignment 2

(Due Wednesday, July 6)
Section 1.3: Vector Equations
  • Reading Goals: Understand algebraic and graphical representation of vectors; know how to convert from vector equations to linear systems of equations to augmented matrices; understand linear combinations and the span of a set of vectors.
  • Required Problems: # 11, 24, 26, 32.
  • Recommended Problems: # 9, 12, 13, 21, 23.
Section 1.4: The Matrix Equation Ax = b
  • Reading Goals: Be able to write a vector equation as a matrix equation; determine whether a matrix equation is consistent and list equivalent conditions; know basic properties of matrix-vector products.
  • Required Problems: # 9, 15, 22, 24.
  • Recommended Problems: # 10, 14, 23, 31.
Section 1.5: Solution Sets of Linear Systems
  • Reading Goals: Find solution sets to homogeneous systems of linear equations and relate to nonhomogeneous linear systems; write solution sets in parametric vector form.
  • Required Problems: # 5, 15, 24, 40.
  • Recommended Problems: # 6, 16, 23, 26, 29.


Assignment 3

(Due Wednesday, July 13)
Section 1.7: Linear Independence
  • Reading Goals: Understand linear dependence and independence; understand the relationship between dependence of the columns of a matrix and the number of solutions to the corresponding matrix equation.
  • Required Problems: # 5, 10, 22.
Section 1.8: Introduction to Linear Transformations
  • Reading Goals: Know the properties of a linear transformation; recognize geometric properties of some linear transformations.
  • Required Problems: # 11, 19, 22, 30.
Section 1.9: The Matrix of a Linear Transformation
  • Reading Goals: Produce a matrix equation for a given linear transformation; understand one-to-one and onto transformations and their relationship with the number of solutions.
  • Required Problems: # 7, 24, 35.


Assignment 4

(Due Wednesday, July 20)
Section 2.1: Matrix Operations
  • Reading Goals: Know how to multiply matrices (when the product is defined), properties of matrix multiplication, and the transpose of a matrix.
  • Required Problems: # 2, 16, 25.
Section 2.2: The Inverse of a Matrix
  • Reading Goals: Know how to solve a system of linear equations using the inverse of a matrix, understand the relationship between row operations and elementary matrices, be able to compute the inverse of a matrix (learn the formula for the inverse of a 2-by-2 matrix).
  • Required Problems: # 7, 10.
Section 2.3: Characterizations of Invertible Matrices
  • Reading Goals: Know and be able to apply the Invertible Matrix Theorem.
  • Required Problems: # 2, 8, 12.
Section 3.1: Introduction to Determinants
  • Reading Goals: Understand the relationship between the determinant and invertibility of a matrix, be able to compute the determinant.
  • Required Problems: # 10, 40.


Assignment 5

(Due Wednesday, July 27)
Section 3.2: Properties of Determinants
  • Reading Goals: Understand how row operations change the value of the determinant, know how to quickly computer the determinant of a triangular matrix, simplify the determinant of a product of matrices.
  • Required Problems: #12, 26, 28.
Section 4.1: Vector Spaces and Subspaces
  • Reading Goals: Know the definition of a vector space and subspaces and examples.
  • Required Problems: #8, 12, 20, 24.
Section 4.2: Null Spaces, Column Spaces, Linear Transformations
  • Reading Goals: Be able to computer the null space and column space of a matrix, understand how they are related and why they are important.
  • Required Problems: #5, 15, 26, 29.


Assignment 6

(Due Wednesday, August 3)
Section 4.3: Linearly Independent Sets; Bases
  • Reading Goals: Know the definition of basis and examples; be able to check whether a given set of vectors forms a basis; be able to find a basis for a given space; know how to use the Spanning Set Theorem.
  • Required Problems: #9, 13, 22.
Section 4.4: Coordinate Systems
  • Reading Goals: Understand the Unique Representation Theorem; find the change-of-coordinates matrix from the standard basis of Rn to a vector space with basis B and use it to convert from standard coordinates into B-coordinates and back again.
  • Required Problems: #7, 16, 32.
Section 4.5: Dimension of a Vector Space
  • Reading Goals: Know the definition of dimension; understand the Basis Theorem and how it simplifies verification of the conditions for a basis; be able to find the dimensions of the Null Space and Column Space of a matrix and how they are related.
  • Required Problems: #14, 20, 25.


Assignment 7

(Due Wednesday, August 10)
Section 4.6: Rank
  • Reading Goals: Know the definition of the row space of A and how to find a basis for it; know the relations among the dimensions of Col(A), Row(A), and Nul(A) and the Theorem on Rank.
  • Required Problems: #4, 10, 13, 18.
Section 5.1: Eigenvalues and Eigenvectors
  • Reading Goals: Know what eigenvalues and eigenvectors are and how to find them; be able to compute the basis for the eigenspace of a matrix corresponding to a given eigenvalue.
  • Required Problems: #5, 15, 18, 22.


Assignment 8

(Due Wednesday, August 17)
Section 5.2: The Characteristic Equation
  • Reading Goals: Understand the relationship between the eigenvalues of a matrix and the characteristic polynomial; be able to compute characteristic polynomials; know the definition of similar matrices.
  • Required Problems: #7, 10, 15, 22.
Section 5.3: Diagonalization
  • Reading Goals: Know how the factorization A = PDP-1 can be used to calculate a formula for Ak; be able to diagonalize a matrix.
  • Required Problems: #4, 11.
Section 6.1: Orthogonality, Inner Products, and Length
  • Reading Goals: Know how to compute the inner product of vectors and how it relates to the length; know how to compute the distance between two vectors; know the definition of orthogonality, how to determine when vectors are orthogonal, and how it relates to the Pythagorean theorem.
  • Required Problems: #20, 24.


Assignment 9

(Due Wednesday, August 24)
Section 6.2: Orthogonal Sets
  • Reading Goals: Know properties of orthogonal sets, how to compute orthogonal projections, how to change into a coordinate system with an orthogonal basis via projections, and know properties of orthonormal matrices .
  • Required Problems: #10, 15, 24.
Section 6.3: Orthogonal Projections
  • Reading Goals: Know the orthogonal projection theorem, the best approximation theorem, and how to compute projections using orthogonal matrices.
  • Required Problems: #3, 14, 18, 22.