Math 121A Homework 3 Solutions
1.
1
= i + j + 2k, where the coefficient of i is non-zero, we may choose s1 = i,
2
then Span(x1 , j, k) = R3 . Now repeat: since x2 = 0 = 2x1 k, we may choose s2 = k,
Math 121A Homework 2 Solutions
1. Suppose that f , are differentiable, and R. Since
( f + g)0 = f 0 + g0
and
( f )0 = f 0
we see that V is a subset of F (R, R) closed under addition and scalar multipl
Math 121A Homework 1 Solutions
1. Functions in F (S, R) are equal if they evaluate to the same thing on all values in S. We check:
f (0) = 1 = g (0)
and
f (1) = 3 = g(1) = f = g
Similarly,
( f + g)(0)
Math 423, Answers for homework 1
1.2.1. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) 1.2.2. True, by definition. False (homework). False: take x = 0. Comment: true if x = 0. False: take a = 0. Comment:
Math121A
Name: Signature:
Sample Final Exam
WQ10
Instructions: 1. Check that you have pages 1 through 5 and that none are blank. 2. Do not spend too much time on a particular problem. Work the easier
Answers to your questions 03/10/10: Administrative questions 1. In class today you said that we would not be tested on . Let me clarify this again: There will be no problems on geometric meaning of de
Answers to your questions 03/08/10: Administrative questions 1. It is great that you have lecture notes for us to print out. We can follow you and it makes the class easier. Thanks for the comment. I
If B is a matrix obtained by interchanging any two rows of A, then |B| = -|A|. If B is a matrix obtained by multiplying a row of A by a nonzero scalar k, then |B| = k|A|. If B is a matrix obtained by
If B is a matrix obtained by interchanging any two rows of A, then |B| = -|A|. If B is a matrix obtained by multiplying a row of A by a nonzero scalar k, then |B| = k|A|. If B is a matrix obtained by
Math 121A Homework 5 Solutions
1. For each matrix we perform row operations until we cannot simplify the matrix further. If
we obtain the identity, then we apply the same sequence of row operations to
Math 121A Homework Midterm Prep
1.
(a) Simply compute:
(x + v) + (y + w)
x
v
x + v
0
L
+
=L
=
y
w
y + w
2(x + v) (y + w)
( x + y) + (v + w)
x+y
v+w
= 0 + 0
0
=
(2x y) + (2v w)
2x y
2v w
x
v
Math 121A: Spring 2016 Midterm
Name:
Student Id#:
Total marks = 50 (per question in brackets)
No calculators or other electronic devices
Unless otherwise stated, include all your working for full cred
Spring 2015 Math 121A Homework
Solutions 3
2.2
2.3
2.4
Matrix representation of a linear
transformation
Composition of linear transformations
Invertibility and isomorphisms
1, 2 (a), 4, 5, 9, 13
1, 3,
Spring 2015 Math 121A Homework
Solutions 2
Wed, Apr. 15
1.5
Linear dependence and independence
1.6
Basis and dimension
2 ( a, c, e, g, i ), 6, 15, 16, 18
1 , 2, 3, 14 , 16,
22 , 29, 33
Linear dependen
Spring 2015 Math 121A Homework
Solutions 3
2.5
3.2
2, 3, 4 , 11, 13
2, 5, 6 , 14
Change of coordinate matrix
Rank of a matrix and matrix inverse
Change of coordinate matrix
Problem 2 : (a) P =
a1 b1
a
Spring 2015 Math 121A - Homework 6
4.4
More facts about determinants
4.5
A characterization of determinants
Problems posted on the website
4, 5 , 6 , 9 , 10,
11 , 16, 17, 18
4.4 - More facts about det
Mia/Wm z Rot/[w 5M
_
LV" V mm] W 12 v.3 PYM m 5m yz/J 0; scalars F_
A /in.W/formw(m W V +0 W is :7 WW 73 V~9W sdlskgl'mz,
u Tong) : NWT/u f Ml novev
a) T041) = c700 M W er WWI eel:
0M
Mia/75 / VLW Steef
Vm smes
' ' )4 vwbr spa. Vm yall F is a 501 wivt M Wailing? aa/a/im and Sta/M mat/VIM)
(for x my and GF Yra anal ax We in V) W Mm Mluag
om am? I
(V5,) 3+6 =5+Y Isa 02! 1,5 in l/ (C
Midterm 121A
- Fall 2016
10/21/2016
There are 4 problems worth a total of 200 points. Show your work and
justify your answers. You may leave your answers in numerical form (eg 57').
Calculators are no
Answers to your questions 03/05/10: Administrative questions 1. Are you going to put some extra credit problems on the final? No. 2. Can you post up answers to the isomorphism worksheet? Yes. 3. I app
Answers to your questions 03/03/10: Administrative questions 1. Can we have more homework-like examples of the systems of linear equations? We'll do it next time. Questions for today's lecture 2. Q2 o