Math 3083 Linear Algebra
NAME:
(Please print clearly)
Sixth Quiz (solutions)
3 Mar 2014
1. For each of the following sets of vectors in R3 , determine (i) whether it spans R3 ,
(ii) whether it is independent, and (iii) whether it is a basis for R3 . Show
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Third Quiz (solutions)
Due 10 Feb 2014
1. For each matrix below, state whether it is elementary or not. If it is elementary, state
exactly what elementary row operation was performed upon the identity
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Second Quiz (solutions)
27 Jan 2014
1. For the matrices A =
12
1 1 2
, B = 0 1 and C =
0
12
12
12
31
nd the
following when possible. If a combination is not possible, write not possible. (The
matrix I2
Math 3083 Linear Algebra
NAME:
(Please print clearly)
First Quiz (solutions)
22 Jan 2014
1. Using only elementary row operations convert each of the following matrices to echelon
form. Note: each can be done in at most two steps.
11
3
0 1 1
1 2 0
(a) 1 2
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Eleventh Quiz (solutions)
Due April 23, 2014
1 1 0
The matrix A = 0 2 2 , has three eigenvalues, 1, 2, and 3
0 2 1
1. For each of those eigenvalues of A, nd an eigenvector.
Ans: To get the eigenvector
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Ninth Quiz (solutions)
Due 11 Apr 2014
1. Let the vector space P3 (polynomials of degree less than 3) be given the inner product
dened by p(x), q(x) = p(0)q(0) + p(1)q(1) + p(2)q(2).
Let p(x) = x2 2x a
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Eighth Quiz (with solutions)
Due 7 Apr 2014
The following problems are based in Rn for some values of n. Use the standard scalar product
in answering them.
2
4
1. For the vectors u = 2 and v = 0 , le
MATH 3083 Linear Algebra
NAME:
(Please print clearly)
Third Exam
25 Apr 2014
2
2
1. For the vectors x = 2 and y = 1 in R3 , do the following:
1
4
(a) Find the scalar product of x and y.
Ans: xT y = 4 2 + 4 = 6
(b) Find x and y .
Ans: x = 9 = 3, y = 21
(c)
MATH 3083 Linear Algebra
Homework (solutions)
1. For each of the following vector spaces, (i) state its dimension and (ii) give an example
of a basis for that space.
(b) R32
1
0
0
0
0 1 0 0
Ans: (a) (i) 4. (ii) , , , .
0
0
1
0
0
0
0
1
(a) R4
(c) P7
(b
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Fifth Quiz (solutions)
Due 26 Feb 2014
1. For each of the following sets S , state whether or not it is a subspace of the indicated
vector space. (Just write yes or no.)
x1
(a) The set S = x2 x1 = x
MATH 3083 Linear Algebra
NAME:
(Please print clearly)
First Exam (solutions)
12 Feb 2014
1. For the following system of equations, (a) write down its augmented matrix, then
(b) using only elementary row operations convert the matrix to reduced echelon for
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Second Exam (solutions)
12 Mar 2014
1. For each of the following sets S , state whether or not it is a subspace of R3 . (Just write
Yes or No.)
x1
Ans: Yes.
(a) S = x2 x1 = x2 2x3
x3
a+b
(b) S = a
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Tenth Quiz (solutions)
16 April 2014
4 0 0
For the matrix A = 1 3 1 , do the following.
2 1 1
1. Find the eigenvalues of A. There should be two of them.
Ans: det(A I) = (4 )(2 4 + 4) = 0. Solve to get
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Seventh Quiz (solutions)
19 Mar 2014
1
1
0
.
0 , 1 , 1
1
0
1
1
1
Let C be the basis for R2 given by C =
,
.
1
2
Finally, let L be the linear transformation from R3 to R2 dened by
x1
x1 x2
L x2 =
.
2x1
Math 3083 Linear Algebra
NAME:
(Please print clearly)
Fourth Quiz (solutions)
Due 10 Feb 2014
Find the following determinants.
1
5 9
1. det 0
0
0
8 11 22
Ans: Because of the row of zeros, this determinant is 0.
1
11
2. det 1
1 1 .
8 11 22
Ans: Because two
MATH 3083 Linear Algebra
NAME:
(Please print clearly)
Final Exam (solutions)
5 May 2014
1. For the following system of equations, (a) write down its augmented matrix, then
(b) using only elementary row operations convert the matrix to reduced echelon form