Self-test 3a
1. For what values of the parameter b will the following system
of equations fail to have a unique solution?
(4 pts)
4x by = 2
b2x by = 3
2. Find the characteristic equation, eigenvalues, and corresponding
eigenvectors for the matrix below.
(
Unit 5 Determinants II: Further Properties
Objectives
When you have completed this unit, you should be able to
1. use row reduction to evaluate the determinant of a square matrix.
2. state the way that an elementary operation on a square matrix changes th
Unit 3 Elementary Matrices and the Matrix Inverse
In Unit 1, you studied elementary row operations as a method for solving
systems of linear equations. Recall that the three elementary row operations
are multiplying a row by a nonzero constant, interchang
Unit 8 Introduction to Vector Spaces
Units 8 and 9 introduce you to the study of abstract vector spaces. How well
you do in these units depends on how well you have learned the materials
presented in Units 1-7. Two- and three-dimensional Euclidean space,
Unit 7 Lines and Planes in Three Dimensions
In this unit, we use vector algebra as a tool to study properties of lines and
planes in three-dimensional space. If you recall the discussion on visualizing
linear equations from Unit 1, you know that a single
Unit 6 Vectors in Two and Three Dimensions
Up to this point, you have studied the basics of matrix arithmetic and matrix
algebra as it is used to solve systems of linear equations. Historically, this is
how the study of matrices began. Parallel to this de
Unit 9 General Vector Spaces
In Unit 8, we considered Euclidean n-space. In this unit, we generalize to
arbitrary vector spaces. We do so by defining a vector space as any set of
objects for which operations of addition and multiplication by a scalar are
Unit 10 Linear Programming:
Geometric Method
One of the major applications of linear algebra is the method for maximizing
or minimizing some quantity, such as profit or cost, known as linear
programming. In a linear programming problem, a linear function,
Unit 4 Determinants I:
Cofactor Expansion and Cramers Rule
In this unit, we introduce the concept of the determinant of a square matrix.
The determinant is a function that associates a unique real number with any
square matrix A. This number is denoted de
Unit 1 Systems of Linear Equations
In this unit, we introduce the idea of a system of linear equations, and
present a method, called Gauss-Jordan elimination for solving such
systems. We also introduce the special case of a homogeneous system of
equations
Self-test 2b
1. Compute determinants of the following matrices:
a.
1 0 3
4 1 2
1 2 3
b.
2 0 1 2
2 0 4 2
2 1 1 0
1 1 3 5
2. Compute the adjoint matrix for each of the matrices in question 1 and use it
to compute the inverse of these matrices.
3
Self-test 3b
1. For what values of the parameter b will the following system
of equations fail to have a unique solution?
(4 pts)
3x + 8by = 5
b2x + 2by = 9
2. Find the characteristic equation, eigenvalues, and corresponding
eigenvectors for the matrix be
Self-test 1a
1. Find the solution set for the following system of equations:
(4 pts)
2x1 + 2x2 + x3 = 4
3x1 4x2 x3 = 6
2. Use Gauss-Jordan elimination to put the following matrix into
reduced row-echelon form:
2
3
1
4
0
1
0
3
(4 pts)
4 2
0
1 4
2
0 2
2
0
1
Self-test 2a
1. Compute determinants of the following matrices:
a.
1 2 0
2 0 2
1 1 3
b.
0 2 1
2 0 1
1 2 1
4 1 2
3
1
0
1
2. Compute the adjoint matrix for each of the matrices in question 1 and use it
to compute the inverse of these matrices.
3. Comput
Self-test 1b
1. Find the solution set for the following system of equations:
(4 pts)
x1 + 3x2 x3 = 2
2x1 x2 + 4x3 = 3
2. Use Gauss-Jordan elimination to put the following matrix into
reduced row-echelon form:
3
1
2
6
1
0
0
2
(4 pts)
0
0 2
1 3 4
1
1 3
Self-test 4b
1. Determine if the set of all quadratic polynomials (i.e., polynomials
of the form a + bx + cx2) forms a subspace of the vector space of
all cubic polynomials.
(4 pts)
2. Show that the set of all n n anti-symmetric matrices forms a
vector sp
Self-test 4a
1. Determine if the set of all cubic polynomials (i.e., polynomials
of the form a + bx + cx2 + dx3) forms a vector space when
addition is defined as addition of polynomials, and multiplication
by a scalar is simply multiplication of the polyn
Unit 2 Matrix Arithmetic
In Unit 1, you learned how to solve systems of linear equations by reducing
the augmented matrix of the system to row-echelon or reduced row-echelon
form. Matrices are a powerful tool for studying systems of linear equations,
and
Assignment 1
This assignment, based on the content of Unit 1 of the Study Guide, is worth 3% of
your grade. We recommend that you hand it in after you complete Unit 1. You must
show all of your work in order to obtain full marks. For your convenience, eac