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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 determinant and change of coordinate. All other topics wi
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 appreciate it. Questions on general topics 2. When is 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 adding a multiple of one row of A to another row of A,
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 adding a multiple of one row of A to another row of A,
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 appreciate that you never rush teaching the material. Than
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 on worksheet, if 2x3, max should be 3 by theorem 3.5. Pl
Answers to your questions 03/01/10: Administrative questions 1. On the final, are we going to have to find the rank/inverse for some huge matrix? For inverse, the largest one will be 3x3. For rank, the largest one will be 3xn. For determinant, the largest
The set of solutions is called the solution set of the system. A system of equation is called consistent if it has at least one solution. Otherwise it is called inconsistent.
A system Ax = b of m linear equations in n unknowns is called homogeneous if b =
The set of solutions is called the solution set of the system. A system of equation is called consistent if it has at least one solution. Otherwise it is called inconsistent.
A system Ax = b of m linear equations in n unknowns is called homogeneous if b =
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 problems first. 3. The grading of this exam is based on
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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 not allowed on the exam. This exam has 8 pages (Count
em!
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 credit
Try all parts of every question, even if you can’t d
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
= L
+L
y
w
x
For the null space:
N ( L) x + y = 0 a
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 the augmented
matrix ( A| I ) whereby we will obtain (
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,
(a) Since x1 =
1
2
1
whence Span(x1 , x2 , j) = R3 . It
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 multiplication. Since it is
clearly non-empty (e.g. f = 0 is a
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) = f (0) + g(0) = 2 = h(0) and
( f + g)(1) = f (1) + g(
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: true if a = 0. True. False: m rows, n columns. False.
The rank of any matrix equals the maximum number of its linearly independent columns; that is, the rank of a matrix is the dimension of the subspace generated by its columns.
The rank of any matrix equals the maximum number of its linearly independent columns; that is, the rank of a matrix is the dimension of the subspace generated by its columns.
Answers to your questions 02/26/10: Administrative questions 1. How many questions will next worksheet have? I don't know yet. 2. How are worksheets dropped? By percentage or grades? Percentage. Questions for today's lecture 3. To find 0 , you can do row
Answers to your questions 02/08/10: Administrative questions 1. Will worksheets always be TF questions? Yes. 2. Can we take this paper home to write questions that come up while studying? Yes. Good idea. 3. Can we go over more examples in class? I will tr