Math 115 Spring 11
Written Homework 10 Solutions
1. For following limits, state what indeterminate form the limits are in and evaluate the
limits.
3x2 4x 4
x2
2x2 8
(a) lim
0
. Algebraically, we hope to be able to factor the
0
numerator and denominator an
Problem 1 (10 pts)
Let T be a linear transformation, we say that a subspace W is T -invariant,
if T (W ) 2 W . Prove that N (T ) and R(T ) are T -invariant.
2
Problem 2 (10 pts)
Prove using induction that if W is a subspace of a vector space V and
w1 , .,
Problem 2 (10 pts)
Let S be a linearly independant subset of a. vector space V, and let u b:
3 VBth in V that is not in S. Show that S U {1)} is linearly dependent 1
and only if v E span(S).
, M e Wm (v
‘)
' l
‘ "Us :10ch («Jwa \{EEFI VHS
5
43w M,1:.,.,w
Problem 1 (10 pts)
Let T : V ! V and S : V ! V be two linear transformations. Show that
ST is invertible if and only if S and T are invertible.
2
Problem 2 (10 pts)
Let V, h, i be a real inner product space. Let W be a subspace of V . Define
W ? to be the
Problem 1 (10 pts)
A linear transformation is invertible if and only if its determinant is not 0.
If S and T are invertible then both det(S) and det(T ) are not 0 and we have
det(ST ) = det(S)det(T ) 6= 0. Reciprocally if det(ST ) 6= 0 then both both
det(
115A Homework 1- Due Friday, October 1.
Section 1.2 # 7,8,11,12,17,18
Section 1.3 # 1,4,6
Homework should be written neatly and clearly explained. If it requires more than one sheet,
the sheets must be stapled. Your final answer must be clearly boxed. Inc
115A Homework 2- Due Friday, October 9.
Section 1.3 # 10,16,19,25,30
Section 1.4 # 1,2(a)(c),5(a),9,11,14,16
Homework should be written neatly and clearly explained. If it requires more than one sheet,
the sheets must be stapled. Your final answer must be
115A Homework 3- Due Friday, October 16.
Section 1.6 # 1,4,11,17,20,26,30
Section 2.1 # 1,3,4,8
Homework should be written neatly and clearly explained. If it requires more than one sheet,
the sheets must be stapled. Your final answer must be clearly boxe
115A Homework 4- Due Friday, October 23.
Section 2.1 # 10,13,15,17,20,22,25,29
Section 2.2 # 1,3,5,8
Homework should be written neatly and clearly explained. If it requires more than one sheet,
the sheets must be stapled. Your final answer must be clearly
Math 115A HW #5
Due May 8, 2015
Section 2.5, Problems 1, 3(a), 3(c), 4, 7 .
Additional Problem: Let V = R2 , and let = cfw_(1, 1), (1, 1) and = cfw_(1, 1), (1, 2). Find the change
of coordinate matrix Q from to , so that
Q[v] = [v] .
Section 4.4, Problems
Jeffrey Wong
Math 151A
Homework 2 Solutions
Winter 2016
Problem 1: Let 0 < q < p and suppose n = + O(n p ). Then there is a constant K > 0 such that
|n | Kn p .
Since q < p and n 1 we have n p nq so
|n | Kn p Knq
i.e. n = + O(nq ) as well. This shows that
Jeffrey Wong
Math 151A
Homework 5 (Midterm Review) Solutions
Winter 2016
Problem 1:
a) The function f is continuous wiht f (0) = 1 and f ( ) = so by the intermediate value
theorem it has at least one zero in [0, ].
b) Set p0 = /2 and let the initial inter
Final Exam
Math 115A
Name:
TA:
Read all of the following information before starting the exam:
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take off points if I cannot see how you arrived at your answer (eve
Jeffrey Wong
Math 151A
Homework 4 Solutions
Winter 2016
Problem 1: The proof here is reminiscent of the derivation of Newtons method itself, but keeping
track of the error term. Because f C2 [ a, b], we can expand f in a Taylor series around the point
pn
Math 115A, Practice Final 1 Solutions
Ian Coley
December 6, 2014
Problem 1.
For which values of the constant c is (1, c, c2 ) a linear combination of (1, 2, 4) and (1, 3, 9)?
Solution.
If we assume that it is a linear combination, we have two scalars a, b
Jeffrey Wong
Math 151A
Homework 3 Solutions
Winter 2016
Problem 1: Let g( x ) = 31 x +
2A
3x
with A > 0. Assume x > 0. Since
2A
1
| g0 ( x )| = |1 2 |
3
x
we have the bound
r
0
| g ( x )| < 1
for x >
A
.
2
To apply the fixed point theorem, we need to show
Problem 1 (10 pts)
TRUE OR FALSE. Each correct answer worth 2 points and each wrong
answer worth -2 points. The minimum for this exercise is 0 points.
1. Two similar matrices have the same eigenvalues.
2. If A is a diagonalizable matrix of size n then A h
1: INTRODUCTION, FIELDS, VECTOR SPACES, BASES
STEVEN HEILMAN
Abstract. These notes are mostly copied from those of T. Tao from 2002, available here
Contents
1.
3.
4.
5.
6.
7.
8.
Introductory Remarks
Fields and Vector Spaces
Three Fundamental Motivations f
Practice Midterm
1
Problem 1. Let V = cfw_(a1 , a2 ) : a1 , a2 I . Dene addition of elements R of V coordinatewise, and for (a1 , a2 ) in V and c I dene R, c(a1 , a2 ) = (0, 0), if c = 0 (ca1 , ac2 ), if c = 0
Is V a vector space over I with these operati
Math 115 Spring 2011
Written Homework 8
Due Friday, March 18
1. For each function, determine whether the function is a polynomial function or not. It it
is not a polynomial function, explain why not. If it is a polynomial function, determine the
leading c
Math 115 Spring 2011
Written Homework 11 Solutions
1. Use properties of exponents to write the following functions in the form 2kx for a suitable
constant k.
(a) f (x) := 45x/2
Solution:
5x
f (x) := 45x/2 = (22 )5x/2 = (2)2 2 = 25x .
(b) g(x) := (24x 2x )
Math 115 Fall 2010
Written Homework 1 Solutions
Due Wednesday, September 1
1. Read the course information. For problem 1, please sign your name signifying that you
have read the information and that you understand that you are responsible for knowing
and
Math 115 Spring 11
Written Homework 12 Solutions
1. Consider f (x) := (x 2)2 (x + 2)2 .
(a) What is the domain of f ? Justify your answer.
Solution: Since f is a polynomial, we know that the domain of f is all of R.
(b) What is the range of f ?
Solution:
Math 115 Spring 2011
Written Homework 7 Solutions
1. Given a = 2 + 3 and b = 2 3. Find each of the following numbers and state whether
it is rational or irrational.
(a) a + b
Solution:
a + b = (2 + 3) + (2 3)
= 2+ 3+2 3
= 4 a rational number
(b) ab
Soluti
Math 115 Spring 2011
Written Homework 9 Solutions
1. Show that the following sequences all limit to 4. Justify your answers.
(a) a(n) := 4 + 1/n
Solution: We know that as n , 1/n > 0 always and 1/n 0. Hence, a(n) > 4 always
and a(n) 4 + 0 = 4 as n .
(b) b
Math 115 Spring 2011
Written Homework 6 Solutions
1. Assume the following series are either arithmetic or geometric. (i) Determine the generating function for the associated sequence. (ii)Determine the limit of the associated sequence.
(iii) Evaluate the
Math 115 Spring 11
Written Homework 2 Solutions
1. Consider the sequences dened by the following functions. For each one (i) nd the rst
5 terms, the 500th term, and the 5000th term of the sequence and (ii) determine the limit
of each sequence and state wh
Math 115 Spring 11
Written Homework 4 Solutions
1. For each arithmetic sequence, nd a function a(n) that describes the sequence and determine the limit of each sequence.
7
1
(a) 5, , 2, , 1, .
2
2
Solution: The generating function for an arithmetic sequen