Homework Set 1 for MAS 4105
The following problems are due on Friday, January 20:
1. Determine whether the set V = R2 with the indicated operations is
a vector space over R.
( x1 , x2 ) + ( y1 , y2 ) = ( x1 + y1 , x2 + y2 )
c(x1 , x2 ) = (cx1 , 0)
If V is
MAS 4105Practice Problems for Exam #1
1. Find a basis for each vector space V . (No justication needed for this problem.)
(a) V = M22 (F )
(b) V = cfw_(x1 , x2 , x3 ) R3 : x1 + x2 5x3 = 0
(c) V = cfw_f P2 (F ) : f (0) = 0
2. In each case determine whether
Answers for Practice Problem Set 1
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(b) cfw_(1, 1, 0), (5, 0, 1)
(c) cfw_X, X 2
1. (a)
2. (a) W is a subspace of R3 .
(b) W is not a subspace of R3 , since (0, 1, 0) W but (1)(0, 1, 0) = (0, 1, 0) W .
3. (a) S is a basis.
4.
Homework Set 2 for MAS 4105
Due Friday, January 27
1. Use Gaussian elimination to nd all solutions to the following system
of equations.
x1 x2 x3 + x4 = 2
x1 + x2 + x3 + x4 = 0
x1
2x 3
=2
2. In each case determine whether y Span(S ).
(a) V = R3 ,
S = cf
Homework Set 3 for MAS 4105
Due Wednesday, February 1
1. Let S be a subset of the vector space V and let z Span(S ). Prove
that Span(S cfw_z ) = Span(S ).
2. Determine (with justication) a basis for the subspace W of P3 (R)
dened by
W = cfw_f P3 (R) : f (
Homework Set 4 for MAS 4105
Due Friday, February 10
1. Dene V = cfw_f P3 (R) : f (2) = 0.
(a) Compute dim(V ).
(b) Find subspaces W1 and W2 of V such that dim(W1 ) = 1 and
dim(W2 ) = 2.
(Problem 2 on Homework Set 3 may be helpful here.)
2. Let V be a vect
Homework Set 5 for MAS 4105
Due Friday, February 17
1. Let T : V W be a linear transformation and let v1, . . . , vn be vectors in V such that T (v1), . . . , T (vn) are distinct linearly independent
vectors in W . Prove that v1 , . . . , vn are linearly
MAS 4105: Linear Algebra I
Spring 2012
Instructor: Kevin Keating
Oce: Little 482
Telephone: 3920281, Ext. 266
E-mail: [email protected]
Web Page: http:/www.math.ufl.edu/~keating/4105/
Oce hours: Mondays 7th period, Wednesdays 7th and 8th periods, or by appo