NAME
MAT 342, Quiz 5, Spring 2015
No calculators
1. (5 pts) Give the precise definition of
Span(v1 , v2 , , vk )
Answer:Span(v1 , v2 , , vk ) is the set of all linear combinations of the vectors v1 , v2 , , vk .
2. (5 pts) Let U and V be subspaces of a ve

MAT 342, Quiz 9 Answers, Spring 2015,
1. (8 pts) Consider the points P = (1, 1, 1), Q = (1, 2, 3) and R = (4, 3, 2).
(a) Find a vector that is perpendicular to both P Q and P R
Answer. P Q= (0, 1, 2)T and
P Q P R=
P R= (3, 2, 1)T so that
i j k
0 1 2 = 3

NAME
MAT 342, Quiz 8, Spring 2015
No calculators
1. (6 pts) Consider the linear transformation L : R22 R2 2 defined by
L(A) = 2A tr(A)I,
where tr(A) denotes the trace of A i.e. the sum of the diagonal entries.
(a) Find a basis for the kernel of L.
Answer.

MAT 342, Spring 2015
Quiz 7 Answers
1. (6 pts) Let E be the following ordered basis for the space of antisymmetric 3 3 matrices
(AT = A): R22 :
0 1 0
0 0 1
0
1 1
E := 1 0 0 , 0 0 0 , 1 0 1
0 0 0
1 0 0
1 1 0
and suppose that
Answer:
0
1
M =1
0
M is a

MAT342 - Quiz 4 - Spring 2015
Answers
1. (5 pts) Let A be a nonsingular matrix. Prove that
(
)
det A1 =
Answer.
1
det(A)
det(A1 A) = det(I) = 1
1 = det(A1 A) = det(A1 )det(A),
(
)
and so det A1 = 1/det(A).
2. (5 pts) Use Cramers rule to find the solution

MAT 342 Exam 1, Spring 2015, Answers
1. (10 pts) Let A be a 4 3 matrix and let E be the elementary matrix such that EA is obtained
from A by adding 3 times the second row to the 4th row.
(a) What is E?
1
0
0
0
(b) What is E1 ?
0
1
0
3
1 0
0 1
0 0
0 3

MAT342 Exams - Fall 2014
EXAM 1
1. (5 pts) Give the precise denition of linear combination.
2. ( 10 pts) After performing row reductions on the augmented matrix for a certain system of linear
equations for x1 , x2 , x3 , and x4 , the following matrix is o

MAT 342
Quiz 12, Spring 2015, Answers
1. (4 pts) Complete the following definitions
(a) Let A be a square matrix then is said to be an eigenvalue of A if there exists a
nonzero vector x such that Ax = x.
(b) A square matrix A is said to be diagonalizable

MAT 342, Quiz 10, Spring 2015
Answers.
1. (8 pts) Let S be the subspace in R4 that is spanned by the vectors (3, 1, 3, 1)T and
(1, 1, 3, 3)T . Find a basis for the subspace S . If (x1 , x2 , x3 )T is orthogonal to the
two given equations we have that thei

MAT 342, Spring 2015
Quiz 6 Answers.
1. (3 pts) Complete the following definition: The vectors v1 , v2 , . . . , vn in the vector
space V are said to be linearly independent if whenever c1 , c2 , , cn are scalars and
c1 v2 + c2 v2 + ., +cn vn = 0, then al

MAT 342, Quiz 11, Spring 2015. Answers.
1. (4 pts) The method of least squares is to be used to fit a quadratic function y = a + bx + cx2
to the data:
x 1 2 3 4 5
y 5 4 3 1 0
What is the system of equations for a, b, and c that needs to be treated with th