Giovanni Forni
Homework Assignments
Assignment 1 Due Fri September 9
Read sections I.1, III.1 and III.2.
Problems:
III, 1 : 2, 4;
Lang
I, 1 : 8, 9;
III, 2: 2.
1
MATH 405
MATH 405 Midterm Exam 2 (Wednesday, March 13, 2013)
Please include all of your responses in the booklet provided. If you do not do the
problems in order please clearly label which responses go with each problem. If you use
a named theorem, please state it
Solutions to MATH 405 Assignment 10
1. (8.1.6-not really) I completely misread this initially but here is a dierent result
(It shows all transformations which have the property T (), ). We want to
nd all linear transformations such that if T (a, b) = (c,
Solutions to MATH 405 Assignment 9
1. (7.3.1) There are only 3 possibilities for the minimal polynomial: x, x2 , x3 . If
the minimal polynomial is x then the polynomial is automatically the 0 matrix,
and no non-zero matrix is similar to the zero matrix. I
Solutions to MATH 405 Assignment 8
1. (7.1.1) We want to prove that a non-zero vector is either an eigenvector, or a
cyclic vector. Note: If T is a scalar multiple of the identity then every non-zero
vector is an eigenvector, since if T = cI then T = c.
A
Solutions to MATH 405 Assignment 7
1. (6.6.5) Let f (x) = an xn + + a1 x + a0 . Then f (E ) = an E n + + a1 E + a0 .
Since E is a projection E 2 = E . Inductively E k = E for all k > 0. Then
f (E ) = (an + + a1 )E + a0 I .
2. (6.6.11) Let Bi be a basis of
Solutions to MATH 405 Assignment 6
1. (6.2.7) Let dimF (V ) = n. All we need to show is that we have a basis of
eigenvectors. Let c1 , . . . , cn be the distinct eigenvalues. Let i V be such
that T (i ) = ci i for i = 1, . . . , n. We claim that B = cfw_1
Solutions to MATH 405 Assignment 4
1. (4.2.8) Let f, g F [x] and c F . Then T (cf + g ) = cf + g (h) = cf (h) + g (h)
so this is a linear transformation. Suppose that T (f ) = 0, that means f (h) = 0.
i
i
Let f =
i ci x . Then f (h) =
i ci (h) . Then sinc
Solutions to MATH 405 Assignment 3
1. (3.5.15) Recall that the space of 2 2 matrices has a standard basis
B=
01
00
00
10
,
,
,
00
00
10
01
. Then T is determined by how it acts on this basis. Multiplying each of these
elements by P we have
PB =
P1,1 0
0 P
Solutions to MATH 405 Assignment 2
1. (3.1.13) First show a b: Suppose rg (T ) ker(T ) = cfw_0, and in addition
that T (T ) = 0. Then T ker(T ), also by denition T rg (T ). So
T inrg (T ) ker(T ) = cfw_0; therefore T = 0.
Now for b a: Suppose that if T (T
Solutions to MATH 405 Assignment 1
1. (2.1.5) Clearly (1), (2) are satised. However (3a) is not satised since in general
a b = b a. Also (3b) is not satised, since a (b c) = a b + c = (a b) c
in general. (3c) and (3d) are satised since 0 Rn and a 0 = a, a
Math405, Homework 1
February 10, 2015
The problems are assigned from the text book:
1. Exercise 4 page 39
2. Exercise 8 page 40
Solution.
a) Suppose that f (x), g(x) Ve and a, b R then af (x) + bg(x) Ve
because af (x) + bg(x) = af (x) + bg(x). This proves
Math405, Homework 2
February 15, 2015
1. Find a basis for the vector space Cn over R.
Solution. For k = 1, . . . , n let ek Cn be a vector with 1 in the k-th spot
and zero elsewhere, and let ek Cn be a vector with i in the k-th spot and
zero elsewhere. Th
MATH 405 Spring 2011 Exam 3
Turn o all electronic devices.
Closed book, closed notes.
Problems 1-8 are worth 10 points each. Problem 9 is worth 25 points.
For typographical reasons, for example the transpose of x will be denoted by xtr .
1. Let Q be the q
MATH 405 Spring 2011 Exam 2
Turn o all electronic devices. * THERE ARE QUESTIONS ON BOTH SIDES OF THIS PAPER *
1. Suppose V is a vector space with a positive denite scalar product < , >.
(a) (4 points) How is |v | dened?
(b) (6 points) State the Schwartz
MATH 405 Spring 2011 Exam 1
Turn o all electronic devices.
1. Suppose T : V W is a linear map, and w0 is a vector in W with a unique preimage
under T . Prove that every vector in W has at most one preimage under T .
2. Suppose that T : V W is a linear map
MATH 405 Spring 2011 Exam 2
Turn o all electronic devices. * THERE ARE QUESTIONS ON BOTH SIDES OF THIS PAPER *
1. Suppose V is a vector space with a positive denite scalar product < , >.
(a) (4 points) How is |v | dened?
(b) (6 points) State the Schwartz
MATH 405 Spring 2011 Exam 3
Turn o all electronic devices.
Closed book, closed notes.
Problems 1-8 are worth 10 points each. Problem 9 is worth 25 points.
For typographical reasons, for example the transpose of x will be denoted by xtr .
1. Let Q be the q
MATH 405 Spring 2011 Exam 1
Solutions
1. (10 pts.) Suppose T : V W is a linear map, and w0 is a vector in W with a unique
preimage under T . Prove that every vector in W has at most one preimage under T .
SOLUTION
Let v0 be the unique vector in V such tha
MATH 405 Spring 2011 Exam 3
1. Let Q be the quadratic form dened on R2 by the rule Q(x) = xtr Ax, where x =
xtr is the transpose of x and A =
x1
,
x2
11
. Find the maximum value of Q on the unit
10
circle.
SOLUTION
The
maximum value of Q on the unit circ
MATH 405 Spring 2011 Exam 1
Solutions
1. (10 pts.) Suppose T : V W is a linear map, and w0 is a vector in W with a unique
preimage under T . Prove that every vector in W has at most one preimage under T .
SOLUTION
Let v0 be the unique vector in V such tha