Math 341
Homework #1
(1.1) #3. Solve the system by using elementary row operations on the equations or
on the augmented matrix.
x1 + 7x2 = 4
2x1 9x2 = 2
Solution: Lets work with the corresponding augmented matrix:
1
74
2 9 2
2R 1 + R 2
17
01
4
2
1
0
7
5
4
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Math 341 Midterm 7/20/12
Directions: Read each question carefully and write your answers in the spaces provided. Unless it
is stated otherwise, you must justify all of your answers in a clear and organized fashion to receive
full credit. You are not
Name:
Math 341 Quiz 2 2/1/13, 12:00
Directions: Read each question carefully. Show your work and answers in a clear and organized
fashion to receive full credit. You are not allowed to use a calculator or any notes for this quiz.
1
3
1
1 , 1 , 5 are lin
Name:
Math 341 Quiz 4 3/8/13, 12:00
Directions: Read each question carefully. Show your work and answers in a clear and organized
fashion to receive full credit. You are not allowed to use a calculator or any notes for this quiz.
111
1. (5 points) Let A =
Name:
Math 341 Quiz 3 2/22/13, 12:00
Directions: Read each question carefully. Show your work and answers in a clear and organized
fashion to receive full credit. You are not allowed to use a calculator or any notes for this quiz.
1. (2 points) Suppose A
Math 341
Final Exam Solutions
1. True/False If the statement is true, give a brief explanation; if it is false, provide a
counterexample.
T
F
If v is a solution to Ax = 0, then v + b is a solution to Ax = b.
False: This would only be true if Ab = b, since
Solutions for Quiz 2
1. (1 point each) Suppose A is a matrix which
1
0
A
0
0
has the following RREF:
0080
1 0 5 0
0 1 3 0
0001
Mark each statement True or False. You dont need to justify your answers.
(a) The columns of A span R4 .
Answers:
T
F
(b) The co
6/29/12
Math 341 Quiz 1
Directions: Read each question carefully and write your answers in the spaces provided. You
need to show all of your work in a clear and organized fashion to receive full credit. You are not
allowed to use a calcuiator or any not
Math 341, Solutions to Practice for the Midterm
1. Complete the following denitions:
(a) (Span) Given v1 , . . . , vp Rn , the set Spancfw_v1 , . . . , vp is equal to the set of all linear
combinations of v1 , . . . , vp . In other words,
Spancfw_v1 , .
Solutions for True/False Practice for the Final
Instructions: Determine whether the given statement is true or false. If it is true, explain why. If it is
false, give a counterexample.
1. Any function f : Rn Rm can be represented as multiplication by a ce
Math 341 Midterm Solutions
1. (4 pts) Complete the following denitions:
The map T : Rn Rm is onto if. . .
. . .every y Rm is the image under T of at least one x Rn .
The set cfw_v1 , . . . , vp Rn is linearly dependent if. . .
. . .there exist scalars c1
Solutions to 341 Midterm from 2007
1. (2pts.) Complete the following denition:
A map T : Rn Rm is called a linear transformation if. . .
Solution: . . . T (u + v) = T (u) + T (v) for all u, v Rn and T (cu) = cT (u) for all
u Rn and all c R.
2. (3pts. each
Math 341 Midterm Solutions
1. (4 pts) Complete the following denitions:
The set of vectors cfw_v1 , . . . , vp is linearly independent if. . .
. . . the equation x1 v1 + + xp vp = 0 has only the trivial solution.
An invertible matrix is a matrix A such t
Name:
Math 341 Quiz 1 1/18/13, 12:00
Directions: Read each question carefully. Show your work and answers in a clear and organized
fashion to receive full credit. You are not allowed to use a calculator or any notes for this quiz.
213
1 0 2
1. (1 point ea
Midterm 1
Solutions
1. True/False If the statement is true, give a brief explanation; if it is false, provide a
counterexample.
T
F
If A is m n and the equation Ax = b is consistent for some b, then the
columns of A span Rm .
False. If A =
1
0
0
,b=
0
1
,
Midterm 1 (from last year)
1. True/False If the statement is true, give a brief explanation; if it is false, provide a
counterexample.
T
F
If A is m n and the equation Ax = b is consistent for some b, then the
columns of A span Rm .
T
F
If the augmented m
Math 341
Midterm Solutions
1. True/False If the statement is true, give a brief explanation; if it is false, provide a
counterexample.
T
F
The equation Ax = b is consistent if the augmented matrix A b has a
pivot position in every row.
False: Such a matri
Midterm 1
Solutions
1. True/False If the statement is true, give a brief explanation; if it is false, provide a
counterexample.
T
F
If A is m n and the equation Ax = b is consistent for some b, then the
columns of A span Rm .
False. If A =
1
0
0
,b=
0
1
,
Math 341
(2.3) #3.
5
7
9
Homework #7
Determine if the matrix is invertible.
03
0 2
01
Solution: This matrix is not invertible. Line (e) of the Invertible Matrix Theorem says
a matrix is invertible if and only if its columns form a linearly independent set
Math 341
Homework #5
(1.7) #3. With T dened by T (x) = Ax, nd an x whose image under T is b, and
determine if x is unique.
1 0 1
0
A = 3 1 5 , b = 5
6
4 2 1
Solution: We are to solve the equation Ax = b, so we row reduce the corresponding
augmented matri
Math 341
Homework #4
(1.5) #8. Describe all solution to Ax = 0 in parametric vector form, where A is row
equivalent to
1 6 0 8 1 2
6
0 0 1 3 4
000 0
0
1
000 0
0
0
Solution: Row equivalent systems (and hence matrices) have the same solution sets, so
lets
Math 341
Homework #3
(1.3) #9. Write a vector equation that is equivalent to the system of equations:
2x1 x2 + 5x3 = 3
x1 8x2 + 2x3 = 5
4x2 4x3 = 5
Solution: This is:
2
1
5
3
1 + x2 8 + x3 2 = 5
x1
0
4
4
5
If you want to be fancy, you can say: x1 a1 +
Math 341
Final Exam Solutions
1. True/False If the statement is true, give a brief explanation; if it is false, provide a
counterexample.
T
F
If v is a solution to Ax = 0, then v + b is a solution to Ax = b.
False: This would only be true if Ab = b, since