Homework 1 Solutions
2. Write the zero vector of M34 (F )
0 0 0 0
0 0 0 0
0 0 0 0
13. Let V denote the set of ordered pairs of real numbers. If (a1 , a2 ) and
(b1 , b2 ) are elements of V and c R, dene
(a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 b2 ) and c(
Homework 3 Solutions
SECTION 1.5
10. Give an example of three linearly dependent vectors in R3 such that
none of the three is a multiple of the other.
1
0
1
0 , 1 , 1
1
0
1
15. Let S = cfw_u1 , u2 , ., un be a nite set of vectors. Prove that S is lin
Homework 5 Solutions
SECTION 2.1
16. Let T : P (R) P (R) be dened by T (f (x) = f (x). T is linear.
Let f P (R).
Then f (x) = an xn + + a1 x + a0 .
Let g(x) =
an
x
n+1 n+1
+ +
a1
2
+ a0 x.
Note that T (g) = g (x) = f (x). So f R(T ). Moreover R(T ) P (R)
Math 235: Linear Algebra
Midterm Exam 1
February 25, 2014
NAME (please print legibly): mill/firms?
Student ID Number:
CIRCLE YOUR INSTRUCTOR: Fatima Mahmood Geordie Richards
0 Read all instructions and all problems carefully.
o This is a closedbook an
MTH 235: Linear Algebra
Midterm 1
0 13
i) Lwbw
Indicate your instructor with a check in the box:
February
NAME (please print legibly):
Your University ID Number:
Giorgis Petridis
M ark Herman
c There are no notes, textbooks, etc. allowe
MTH 235 - Homework 3 Solutions
Question 1. This is done by modifying the proof of Theorem 1.9 as done in Example 6.
The set cfw_v1 , v2 is linearly independent as the vectors are not multiple of each other. Adding v3
gives rise to a linearly dependent se
MTH 235 - Homework 9 Sections 4.1 - 4.3
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. You are gi
MTH 235 - Homework 4 Section 2.1
Question 1.
Discussion: We know how T acts on (1, 2) and (1, 3), which incidentally form a basis for R2 , so we
express (1, 0) as a linear combination of the two vectors and use the linearity of T . For the second
part we
MATH 235
Final ANSWERS
May 5, 2015
1. (10 points)
Fix positive integers m, n and consider the vector space V of all m n matrices with entries
in the real numbers R.
(a) Find the dimension of V and prove your answer. Please carry out all the steps of your
MTH 235 - Homework 8 Section 3.3 & 3.4
Question 1. True of False?
Solution: (a) True Since Ax = 0 for A Mmn always has the trivial solution x = 0 Rn .
(b) False Consider the system of 2 equations in 3 unknowns x z = 0, y z = 0 which has
infinitely many so
MTH 235 - Homework 11 Sections 5.1, 5.2, 5.4
Question 1. True of False? For each of the following, decide if the statement is true or false.
Provide brief justification for your answer (you do not have to give full-blown proofs, brief explanations will su
MTH 235 - Homework 7 Sections 2.5, 3.1
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of Fal
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Subspaces
Linear Independence & Dependence
Let V be a vector space of dim n 2. Let W1 and W2 be subspaces
of V s.t W1 = W2 = V . Show that dim(W1 W2 ) dim(V ) 2.
Since W1 , W2 = V , we see that dim(W1 )< V and dim(W2 )< V , as
proved in class and in the b
Vector Spaces
Vector space V over a eld F is a set on which the operations
addition and scalar multiplication, are dened so that for all
x, y V and for all a F there exists a unique element x + y V
and exists the unique element ax V such that the followin
Math 235: Linear Algebra
Midterm Exam 2
November 19, 2013
NAME (please print legibly):
Your University ID Number:
Please circle your professors name:
Friedmann
Tucker
The presence of calculators, cell phones, iPods and other electronic devices
at this ex
Math 173Q
Practice Second Midterm
1.
(a) Let c, a1 , . . . , an be real numbers. True or false and explain: if there
are any real numbers (t1 , . . . , tn ) such that
a1 t1 + + an tn = 0
then there is a solution to the equation
a1 x1 + + an xn = c.
(b) Le
Practice Questions
1. Let T : R2 R2 be linear and suppose T 2 = T0 , where T0 denotes the zero map.
(a) Show that R(T ) = R(T 2 ).
Solution: Since T 2 = T0 and R(T 2 ) R(T ) we know 1 rank(T 2 ) rank(T ). From
this it immediately follows that if rank(T )
Review Problems for the Final Exam
The in-class portion of the exam will cover through Chapter 5 of the textbook, including eigenvalues, eigenvectors, and diagonalization, but not inner
product spaces.
1. Let be the standard basis of P3 (R).
(a) Is there
Practice Questions
1. Let T : R2 R2 be linear and suppose T 2 = T0 , where T0 denotes the zero map.
(a) Show that R(T ) = R(T 2 ).
(b) Show that T 3 = T0 .
2. (a) Find a matrix A such that
A
2
1
=
1
0
and
A
3
2
=
0
1
Is there another matrix satisfying thi
MTH 235 - Homework 12 Chapter 6
Question 1. Let V be an inner product space.
(a) Prove that kx + yk2 + kx yk2 = 2kxk2 + 2kyk2 holds for all x, y V.
Solution:
kx + yk2 + kx yk2 = hx + y, x + yi + hx y, x yi
= hx, xi + hx, yi + hy, xi + hy, yi + hx, xi hx,
MTH 235 - Homework 9 Chapter 4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. You are given two p
MTH 235 - Homework 6 Section 2.4-2.5
Question 1. True of False?
Solution: (a) True - by definition of invertibility.
(b) True - since IV IV = IV this follows from the definition of inverse map.
(c) False - If Onn is the n n zero matrix, then Onn A = Onn =
MTH 235 - Homework 6 Section 2.4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of False? Fo
MATH 235
Midterm ANSWERS
April 14, 2015
1. (16 points)
(a) Find two disjoint finite spanning sets for the following subspace of R3 :
W = cfw_(x, y, z) R3 : x + y + z = 0.
Recall that two sets are disjoint when they have no element in common. Show all your