Math 420, Exam 1, 15 Feb. 2012.
Solutions:
Problem 1 (2 points) (a) Dene: basis of a vector space.
(b) Dene: dimension of a vector space.
Solution. (a) a linearly independent set which spans the vector space;
(b) the number of vectors in any basis (I am r
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Math 420
Homework 6
2/17/12
3.4.10 We have seen that the linear operator T on R2 dened by T ( x1 , x2 ) = ( x1 , 0) is
represented in the standard ordered basis by the matrix
A=
10
.
00
This operator satises T 2 = T . Prove that if S is
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Math 420
Homework 5
2/10/12
3.1.3 Describe the range and the null space for the differentiation transformation of
Example 2. Do the same for the integration transformation of Example 5.
For the differentiation transformation of Example
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Math 420
Homework 4
2/3/12
2.6.2 Let
1 = (1, 1, 2, 1),
2 = (3, 0, 4, 1),
3 = (1, 2, 5, 2).
= (3, 1, 4, 4),
= (1, 1, 0, 1).
Let
= (4, 5, 9, 7),
(a) Which of the vectors , , are in the subspace of R4 spanned by the i ?
(b) Which of the
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Math 420
Homework 3
1/27/12
2.2.3 Is the vector (3, 1, 0, 1) in the subspace of R4 spanned by the vectors (2, 1, 3, 2),
(1, 1, 1, 3), and (1, 1, 9, 5)?
By Theorem 3, this is true if and only if there exist scalars c1 , c2 , and c3 in R
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Math 420
Homework 2
1/20/12
1.5.8 Let
C=
C11 C12
C21 C22
be a 2 2 matrix. We inquire when it is possible to nd 2 2 matrices A and B such that
C = AB BA. Prove that such matrices can be found if and only if C11 + C22 = 0.
It is easy to s
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Math 420
Homework 1
1/13/12
1.2.6 Prove that if two homogeneous systems of linear equations in two unknowns have
the same solutions, then they are equivalent.
Note that adding the trivial equation 0x + 0y = 0 does not affect equivalence
Math 420 Quiz 5, 11 April, 2012. Solutions
010
Problem 1 (2 points) Here is a real matrix: F := 0 0 2. Give the
400
characteristic polynomial, minimal polynomial, the rational canonical form.
Decide whether there is a cyclic vector.
Solution. The characte
S cores
1:
2:
3:
T otal :
/2
/2
/1
/5
Math 420, Quiz 3, 7 Mar., 2012.
Solutions
NOTICE: Give justications for your
statements or you may not get credit.
Problem 1 (2 points) Let V be a nite dimensional vector space and let
V be the dual space. The annihil
Math 420, Quiz 1, 1 Feb. 2012. Solutions:
Problem 1 (2 points) You are given a vector space V of dimension 6 and
subspaces U1 , U2 such that dim(U1 ) = 3 and dim(U2 ) = 4. Give your best
description of possible values of dim(U1 + U2 ) and dim(U1 U2 ).
Sol
Math 420, Quiz 1, 20 Jan. 2012. Solutions:
Problem 1 (1 points) Let R be a ring. (i) Prove that if a R, there is
only one element x R so that a + x = 0.
(ii) Prove that in R, (1)a = a (a means the unique element of R
which satises a + (a) = 0).
Solution.
Math 420, Exam 2, 28 March, 2012.
Solutions:
Problem 1 (3 points) Let F2 = cfw_0, 1 be the eld of integers modulo 2.
Dene R := F2 [x], the polynomial algebra.
(1.a) Find the g.c.d. of polynomials f (x) = x5 + x3 + x2 + 1 and g (x) =
x4 + x2 + 1 in R. Give
Nutt 1
Scanner Technology
The first scanner was called a Reading Machine and was developed in 1960 by a Russian born
engineer named Jacob Rabinow.
This first device could scan printed material and then compare each character to a set of standards.
For the