MAT 310
Quiz 9
Name:
April 11, 2013
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1. (5 points) Let Tr : Mat22 (R) R be the map dened by
ab
cd
Tr
= a + d.
Compute the dimension of the kernel of Tr.
Solution: Since dim(Mat22 (R) = 4, the Rank-Nullity Theorem tells us that
MAT 310
Quiz 3
Name:
January 31, 2014
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1. (5 points) (Denition) An indexed set cfw_v1 , v2 , . . . , vp in Rn is said to be linearly independent if
Solution: the vector equation
x1 v 1 + x 2 v 2 + + xp v p = 0
has only the tr
MAT 310
Quiz 4
Name:
February 7, 2014
Academic Integrity Signature:
1. (5 points) (Denition) A transformation T : Rn Rm is said to be linear if
Solution:
T (cv) = cT (v),
for all c R and v Rn
T (u + v) = T (u) = T (v),
for all u, v Rn
2. (5 points) (Sta
MAT 310
Quiz 1
Name:
January 17, 2014
Academic Integrity Signature:
1. (6 points) What are the three elementary row operations? Give the names and a brief
description.
Solution:
1. (replacement): Replace an equation with the sum of itself and a scalar mul
MAT 310
Quiz 2
Name:
January 24, 2014
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1. (4 points) (Denition) Let S = cfw_v1 , v2 , , vn be a nonempty set of vectors. The span
of S, denoted Span(S), is
Solution: the set of all linear combinations of vectors from S; that
MAT 310
Name:
Quiz 6
February 28, 2014
Academic Integrity Signature:
1. (5 points) (Denition) For n 2, the determinant of an n n matrix A = aij is
Solution:
n
(1)1+j a1j det(A1j ),
det(A) =
j=1
were A1j is the submatrix of A formed by deleting the 1st row
MAT 310
Name:
Quiz 7
March 17, 2014
Academic Integrity Signature:
1. (3 points) Give the properties dening a subspace H V of a vector space V . Be
precise. e.g. Next to H is closed under addition, you must write out precisely what
this means to get full c
MAT 310
Name:
Quiz 8
March 28, 2014
Academic Integrity Signature:
1. (3 points) (Denition) The rank of a matrix A is
Solution: the dimension of Col(A).
2. (4 points) Answer each question by circling True if it must be true and False if it is ever
false. N
MAT 310
Quiz 9
Name:
April 4, 2014
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1. (3 points) (Denition) Let A be an n n matrix. A scalar is an eigenvalue of A if
Solution: (A I)x = 0 has a nontrivial solution.
2. (3 points) (Denition) Let A be an n n matrix. A nonzero
MAT 310
Quiz 10
Name:
April 11, 2014
Academic Integrity Signature:
4 2
. Find an invertible matrix P and a diagonal matrix D
1
5
such that A = P DP 1 .
1. (5 points) Let A =
Solution: The characteristic polynomial is
det(A I) = (4 )(5 ) 2 = 2 9 + 18 = ( 6
MAT 310
Test 2
Name:
March 21, 2014 (noon12:50 PM)
Academic Integrity Signature:
I have abided by the UNCG Academic Integrity Policy.
Read all of the following information before starting the exam:
It is to your advantage to answer ALL of the 16 question
MAT 310
Test 1
Name:
February 17, 2014 (noon12:50 PM)
Academic Integrity Signature:
I have abided by the UNCG Academic Integrity Policy.
Read all of the following information before starting the exam:
It is to your advantage to answer ALL of the 19 quest
MAT 310
Name:
Quiz 5
February 21, 2014
Academic Integrity Signature:
1. (10 points) Complete the statement of the Invertible Matrix Theorem. Two statements
are lled in for you. Give the remaining ten in complete sentences.
Let A be an n n matrix. Then the
MAT 310
Quiz 8
Name:
March 28, 2013
Academic Integrity Signature:
1. (3 points) Complete the statement of the Unique Representation Theorem. Let B =
cfw_b1 , . . . , bn be a basis of for a vector space V . Then for each x V , there exists
Solution: a uni
MAT 310
Quiz 10
Name:
April 18, 2013
Academic Integrity Signature:
1. (4 points) Let A be an n n matrix. Complete the sentence to dene the terms eigenvalue and eigenvector.
A vector v = 0 is said to be an eigenvector of A corresponding to the eigenvalue i
MAT 310
Name:
Quiz 1
January 17, 2012
Academic Integrity Signature:
1. (5 points) A linear equation in variables x1 , x2 , . . . , xn is an equation that can be written
in the form
Solution:
a1 x1 + a2 x2 + + an xn = b,
where a1 , a2 , . . . , an , b R.
2
MAT 310
Quiz 7
Name:
March 14, 2013
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1. (5 points) Dene linear transformation.
A map T : V W between vectors spaces V and W is a linear transformation if
Solution:
1. T (u + v ) = T (u) = T (v )
2. T (cv ) = cT (v )
for all u,
MAT 310
Name:
Quiz 2
January 24, 2012
Academic Integrity Signature:
1. (5 points) Give the three properties a matrix must satisfy if it is in echelon form. Then
give the additional two properties it must satisfy if it is in reduced echelon form.
Solution:
MAT 310
Quiz 4
Name:
February 14, 2012
Academic Integrity Signature:
1. (5 points) If A is a m n matrix, and if B is an n p, then the product P = AB is the
matrix dened by (give columns of P or (i, j )-entry pi,j )
Solution: The product P = AB is a m p ma
MAT 310
Name:
Quiz 3
January 31, 2012
Academic Integrity Signature:
1. (5 points) Let S = cfw_v1 , v2 , , vn be a nonempty set of vectors. The span of S , denoted
Span(S ), is
Solution: the set of all linear combinations of vectors from S ; that is
Span(
MAT 310
Test 2
Name:
February 28, 2013 (12:301:15 PM)
Academic Integrity Signature:
I have abided by the UNCG Academic Integrity Policy.
Read all of the following information before starting the exam:
It is to your advantage to answer ALL of the 23 quest
MAT 310
Test 3
Name:
April 4, 2013 (12:301:15 PM)
Academic Integrity Signature:
I have abided by the UNCG Academic Integrity Policy.
Read all of the following information before starting the exam:
It is to your advantage to answer ALL of the 21 questions
MAT 310
Test 1
Name:
February 7, 2013 (12:301:15 PM)
Academic Integrity Signature:
I have abided by the UNCG Academic Integrity Policy.
Read all of the following information before starting the exam:
It is to your advantage to answer ALL of the 20 questi
MAT 310
Quiz 6
Name:
March 7, 2013
Academic Integrity Signature:
1. (5 points) Give the ten axioms that dene a vector space.
A vector space V is a nonempty set of objects, called vectors, on which are dened two
operations, called addition and multiplicati
MAT 310
Name:
Quiz 5
February 21, 2012
Academic Integrity Signature:
1. (10 points) Complete the statement of the Invertible Matrix Theorem. Two statements
are lled in for you. Give the remaining ten.
Let A be an n n matrix. Then the following statements
MAT 310
Name:
Test 3
April 23, 2014 (noon12:50 PM)
Academic Integrity Signature:
I have abided by the UNCG Academic Integrity Policy.
Read all of the following information before starting the exam:
It is to your advantage to answer ALL of the 13 question