MA 2073
Homework 5
Assigned 2/12/2010
Due 2/16/2010
1. Consider the vector space V = spancfw_ex , ex , i.e., the space consisting of all linear
combinations aex + bex for arbitrary scalars a and b.
a. (10 points) Show that ex and ex are linearly independe
MA 2073
FINAL EXAM SOLUTIONS
March 5, 2010
In working the following exam, you may use a calculator but no books or notes.
Show your work on all but the rst problem. Use the backs of the pages if you need
more space, but please make clear which problems th
MA 2073
MID-TERM EXAM SOLUTIONS
February 8, 2010
In working the following exam, you may use a calculator but no books or notes.
Show your work on all but the rst problem. Use the backs of the pages if you need
more space, but please make clear which probl
Mid-Term Preparation Material
Things to Know:
Know the material discussed in class and the related material covered in the text (of
course). This should be obvious but is explicitly stated here as a reminder that the
exam will not be restricted to just t
Solving Simple Systems of Linear Equations with Gaussian Elimination
The following illustrates the procedures and principles involved in solving simple systems of linear equations using Gaussian elimination followed by back substitution.
First, consider t
Section 2.4 Practice Problems
MA 2073
Problem Set 2.4, problems 15, 8, 12, 18,22, 29, 32
These problems are intended to be helpful in preparing for the mid-term exam. Writeups
of the problems are not to be turned in.
Remarks:
In the following, dim means
MA 2073
Homework 1
January 15, 2010
Terminology: Naive Gaussian elimination refers to Gaussian elimination as discussed so far in class, i.e., Gaussian elimination with no interchanging of equations or
unknowns (or, equivalently, interchanging of matrix r
MA 2073
Homework 2
Assigned 1/22/2010
Due 1/26/2010
1. (10 points) Solve each of the linear systems below using Gaussian elimination with
partial pivoting followed by back substitution, as follows: (i) Put the system in matrixvector form Ax = b. (ii) Perf
MA 2073
Homework 6
Assigned 2/16/2010
Due 2/19/2010
x
denotes the vector in I 2 with
R
y
components x and y , and (x, y ) denotes the point in the plane with coordinates x and
x
y . Of course, there is a natural identication of
with (x, y ), and I sometim
Homework 3
MA 2073
Assigned 1/28/2010
Due 2/1/2010
1. (10 points) Show that if S and T are subspaces of a vector space V , then S T is
also a subspace.
2. (10 points) Describe the column spaces (lines or planes) of the following matrices:
1
A = 0
0
2
0,
0
Homework 8
MA 2073
Assigned 3/2/2010
Due 2/5/2010
0
1
R
1. Consider S = span 1 , 1 , a two-dimensional subspace of I 3 .
1
0
a. (10 points) Find an orthonormal basis of S .
1
b. (10 points) Find the projection of b = 1 onto S .
1
1
2. Consider A = 1
0
1
MA 2073
Homework 7
Assigned 2/23/2010
Due 2/26/2010
1. (5 points each) Find S for each of the following subspaces S . In each case, give the
dimension of S and S .
v
a. S = v = 1 : v2 = v1 .
v2
v1
b. S = v = v2 : v2 = 0 and v3 = v1 .
v3
2. (10 points) App
Final Exam Preparation Material
Things to Know:
Know the material discussed in class and in the handouts, together with the related
material covered in the text. This familiar admonition should be obvious but is
explicitly stated here as a reminder that