Selected Solutions
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 t
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 vecto
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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
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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 Exampl
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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
Selected Solutions
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
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 w
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 dimensiona
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 o
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 uni
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
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 the