Math 2S3, Test 2
March 13, 2015
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full credit. Plea
Math 2S3, Test 2
March 13, 2015
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full credit. Plea
Math 2S3, Test 1
Bradd Hart, Feb. 6, 2015
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full cr
Assignment 3, Math 2S3
Due Feb. 25 in class
(1) Show that if A is an n n matrix over K then
cfw_f K[x] : f (A) = 0
is an ideal. Conclude that the generator of this ideal is the
minimal polynomial of A and it divides the characteristic polynomial of A.
(2)
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
DUE :
H OMEWORK 3
F RIDAY 23 J ANUARY 2009
Required problems (to be handed in):
For all problems below, let V be a nite-dimensional complex vector space, and let , denote a hermitian inner product , : V V .
1
Assignment 5, Math 2S3
Due Apr. 7, in class
1. (a) Suppose that N is the n n matrix with 1s above the diagonal
and zeroes everywhere else. Compute eN .
(b) Suppose that A = I + N - a Jordan block. Solve the system of
linear dierential equations y = Ay by
Assignment 4, Math 2S3
Due Mar. 11 in class
(1) In Chapter XI of Langs book, the rst section is entitled The
Euclidean Algorithm but the main theorem he proves is really
just long division; lets actually prove the Euclidean algorithm:
suppose that f, g K[
Assignment 2, Math 2S3
Due Feb. 4 in class
(1) (a) How many elements are in an n-dimensional vector space
over the eld with 2 elements, Z2 ?
(b) How many invertible 2 x 2 matrices are there over Z2 ? How
many invertible 3 x 3 matrices are there over Z2 ?
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
DUE :
H OMEWORK 2
F RIDAY 16 J ANUARY 2009
Required problems (to be handed in):
1. Find a general formula for the quotient of two complex numbers. In other words, if a + bi, c + di are
two complex numbers and
Assignment 1, Math 2S3
Due January 20, in class
1. In class we showed that in any group, the inverse is unique. If (G, ) is
a group and we write u for the inverse of u, show that (u) = u for
all u G.
2. Give an example of a non-commutative semi-group whic
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
DUE
H OMEWORK 6
F RIDAY 27 F EBRUARY 2009
This sequence of Required/Recommended Problems analyzes in detail the adjoint representation of su(2)
on su(2). That is just a fancy way of saying that the 2 2 matric
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
DUE
H OMEWORK 7
F RIDAY 6 M ARCH F EBRUARY 2009
R EQUIRED PROBLEMS
(1) Let V = C be the standard complex n-dimensional vector space equipped with the standard hermitian inner product. Let W V be a k-dimension
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
DUE :
H OMEWORK 10
F RIDAY 3 A PRIL 2009
Required problems (to be handed in):
1. What is d(C) (as dened in class for a linear code C) for the ISBN linear code in V (10, 11)?
2. Prove: if C is a p-ary [n, k]-c
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
H OMEWORK 8
20 M ARCH 2009
DUE
Required problems:
All notation and terminology follows the lectures.
1. Recall that Kirkhoffs Current Law states that, for a solution I C1 to the network problem, the
algebraic
Math 2S3, Practice test
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full credit. Please be su
Math 2S3
Dr. Bradd Hart
Apr. 21, 2014
This examination is 3 hours in length. Attempt all 10 questions. The total number of available
points is 50. Marks are indicated next to each question. Write your answers in the booklets
provided. You must show your w
Math 2S3, Test 1
Bradd Hart, Feb. 6, 2015
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full cr
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Assignment 2, Math 2S3
Due Feb. 4 in class
(1) (a) How many elements are in an n-dimensional vector space
over the eld with 2 elements, Z2 ?
(b) How many invertible 2 x 2 matrices are there over Z2 ? How
many invertible 3 x 3 matrices are there over Z2 ?
Assignment 3, Math 2S3
Due Feb. 25 in class
(1) Show that if A is an n n matrix over K then
cfw_f K[x] : f (A) = 0
is an ideal. Conclude that the generator of this ideal is the
minimal polynomial of A and it divides the characteristic polynomial of A.
(2)
Assignment 4, Math 2S3
Due Mar. 11 in class
(1) In Chapter XI of Langs book, the rst section is entitled The
Euclidean Algorithm but the main theorem he proves is really
just long division; lets actually prove the Euclidean algorithm:
suppose that f, g K[
Assignment 5, Math 2S3
Due Apr. 7, in class
1. (a) Suppose that N is the n n matrix with 1s above the diagonal
and zeroes everywhere else. Compute eN .
(b) Suppose that A = I + N - a Jordan block. Solve the system of
linear dierential equations y = Ay by
Assignment 1, Math 2S3
Due January 20, in class
1. In class we showed that in any group, the inverse is unique. If (G, ) is
a group and we write u for the inverse of u, show that (u) = u for
all u G.
2. Give an example of a non-commutative semi-group whic
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
H OMEWORK 9
D UE : 27 M ARCH 2009
Required problems (to be handed in):
1. This problem puts together everything weve discussed about electrical networks, using the theory to
actually solve the network problem