The problems are assigned from the text book:
1. Exercise 4 page 66
Solution.
Since we know dimC C3 = 3, it sufces too show that 1 , 2 , 3 are linearly
independent. Suppose 3 ci i = 0 for ci C. So we
Chapter 1R. Problem lRQ
a Bookmark
L. w _ r_., .u.,. WV,
a+c=d - (3)
Since (1) is passing through C (0, 1,2
b+2c=d (4)
Step 2 of 3 A
From (2) and (3)
d-a=b
And
d~a=b
Implies,
b = c
From (4)
Show all s
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 gener
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 ) = cf
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. Multiply
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.
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
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 ) =
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
Chapter 1R. Problem TRQ
D Bookmark
Step 2 of 2 A
Substitute the value of u. in (1)
(2111)2 1-: =1
2: 4a: + u: = l
:- 5u =1
i
:9 u =
2 5
:> u - i-l
J3
l
u, -2u1 - 2 is
2 2
7? 73
l l
The unit vectors
Chapter 1R. Problem TRQ
Given nisorlhogonalto lzj
3
l
-2:
3
a Bookmark
Substitute the value of u. in (1)
(4:45)2 +31: = I
:> 4u+u=l
=- 5n=l
1
=9 u=
* 5
l
:3 uz=iE
Step 2 of 2 A
Show all steps: C
UK
al
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
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)
Chapter 1R, Problem lRQ
Problem
Find the general equation ol the plane through the points A(1, 1, 0), 3(1, 0, 1), and cm, 1, 2).
Step-by-step solution
Step10f3 A
Consider ageneral equation ol plane,
a
Chapter 1R. Problem SRQ
D Bookmark
Problem
Find the general equation ol the plane through the point (1, 1, 1) that is perpendicular to the line with
parametric equations
x= 2:
y: 3+2t
z=l+t
Step-by-st
Chapter LR. Problem QRQ
a Bookmark
Problem
Find the general equation ol the plane through the point [3, 2, 5) that is parallel to the plane whose
general equation is 2x + 3y? 2 = 0.
Step-by-step solut
Chapter 1R. Problem SRQ
D Bookmark
Step 3 of 3 A
It a plane is passing through the point (xl,yl,zl ] , and il a.b,care the direction ratios of its normal
(perpendicular line to the plane), then the eq
Chapter 1R. Problem SRQ
D Bookmark
z = l H
Here. r is a parameter.
The objective is to lind the general equation 01 the plane through lhe point (l_l,l)the1 is perpendicular
to the above line.
Step 2 o
Chapter 1R. Problem QRQ
Stepinfl A
Let ax+by+cz=d .(I)
Be the required plane equation passing lhrough P = (3, 2. 5) and parallel to the plane
2x+3y-z=0 "(2)
Since (1) a (2) are parallel both have the
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,
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
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 ea
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
ci
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