## Math 22

### Homework

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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.