Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
September 11, 2014
1/8
Plan for today
Review of last weeks materi
Math 140 - Fall 2017 - Homework - Chapter 7 and 8
Give complete, well written solutions to the following exercises:
1 0 1 0
1. (10 points) Let A =
.
0 1 0 1
(a) Find the singular value decomposition,
Math 140 - Fall 2017 - Homework - Chapter 6
Give complete, well written solutions to the following exercises:
1 0
1 2
1. (5 points) Let A =
,B =
.
1 1
0 1
(a) Are the eigenvalues of AB equal to the pr
Math 140 Linear Algebra Fall 2017 Class Starter 2.1, 2.2
Name: K521 , N Number:
Date: Section:
You have 10 minutes to complete this class starter. No calculators. electronic devices, notes or other ou
Math 1.40 -= Linear Algebra n Fall 2017 =- Class Starter 6.1
Name: 43 I N Number:
Date: . Section:
You have 10 minutes to complete this class starter. N0 calculators. electronic devices. notes or ot
Math 140 - Linear Algebra -. Fall 2017 Class Starter 7.1,7.2,7.4
Name: KE z N Number:
Date: Section:
You have 10 minutes to complete this class starter. No calculators, electronic devices, notes or ot
Solutions to Exercises
11
23 If ordinary elimination leads to x C y D 1 and 2y D 3, the original second equation
24
25
26
27
28
29
30
31
32
could be 2y C `.x C y/ D 3 C ` for any `. Then ` will be the
Solutions to Exercises
2
Problem Set 1.1, page 8
1 The combinations give (a) a line in R3
2
3
4
5
6
7
8
9
10
11
(b) a plane in R3 (c) all of R3 .
v C w D .2; 3/ and v ! w D .6; !1/ will be the diagona
Solutions to Exercises
z2 ! z1 D b1
10 z3 ! z2 D b2
0 ! z3 D b3
7
z1 D !b1 ! b2 ! b3
z2 D
!b2 ! b3
z3 D
!b3
"
# " #
!1 !1 !1
b1
0 !1 !1
b2 D !1 b
D
0
0 !1
b3
11 The forward differences of the squares
Solutions to Exercises
21
46 Inverting the identity A.I C BA/ D .I C AB/A gives .I C BA/!1 A!1 D A!1 .I C
AB/!1 . So I CBA and I CAB are both invertible or both singular when A is invertible.
(This re
Solutions to Exercises
46
25 The column space of P will be S . Then r D dimension of S D n.
26 A!1 exists since the rank is r D m. Multiply A2 D A by A!1 to get A D I .
27 If AT Ax D 0 then Ax is in t
Solutions to Exercises
42
23 As in Problem 22: Row space basis .3; 0; 3/; .1; 1; 2/; column space basis .1; 4; 2/,
24
25
26
27
28
29
30
31
32
.2; 5; 7/; the rank of (3 by 2) times (2 by 3) cannot be l
Solutions to Exercises
51
36 Q; R!
! D
" q r.A/ produces from A (m by n of rank n) a full-size square Q D Q1 Q2 !
R
. The columns of Q1 are the orthonormal basis from Gram-Schmidt of the
0
column spac
QUIZ #4
Write your name at the top of the page. Put the answer in the provided box, or
circle the answer. Do not Show any working.
(1) (No partial credit given.)
Suppose A is s 51 X 37 matrix with ran
QUIZ #1
Write your name at the top of the page. Put the answer in the provided box, or ‘
circle the answer. Do not Show any working.
(1) (2 points) In this graphic, u and v axe unit vectors.
y
E
E
E
E
l
t
QUIZ #2
Write your name at the top of the page. Put the answer in the provided box, or
circle the answer. Do not show any working.
(1) (1pt) Compute the matrix product
120
021
tor—lot
n—w—tr—l
(
s
i
1
i
31
i
J
J
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NYU
MATH-Uh. 110
Linear algebra:
Midterm Exam 1
Spring 2016
Name: E UVK CD & bELH/OTYG.
This exam is scheduled for 100 minutes. No calcuiators, notes, or other outside mate—
Math 140 Linear Algebra - Fall 2017 - Class Starter 2.3, 2.4
Name: KE N Number:
Date: Section:
You have 10 minutes to complete this class starter. No calculators, electronic devices, notes or other ou
Math 140 Linear Algebra - Fall 2017 Class Starter 1.3
Name: K52] N Number:
Date: Section:
You have 10 minutes to complete this class starter. No calculators, electronic devices, notes or other outside
Math 140 - Fall 2017 - Homework - Chapter 5
Give complete, well written solutions to the following exercises:
1. (5 points) Use elementary row operations
the following matrices.
0 a 0
0 a 0
0 0 b
A =
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
October 2, 2014
1/8
Plan for today
Quick review of last weeks mat
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
October 23, 2014
1/7
Plan for today
Review of last weeks material
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Mid-term quiz, Oct. 30, 2014
Example solution
1. [2+2+2+2+2pt] Let u1 , u2 , u3 , w be vectors in a linear space V . Are the following
statements
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
December 11, 2014
1 / 18
Organization
Plan for today (our nal cla
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
November 6, 2014
1/8
Plan for today
Review of last weeks material
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
September 18, 2014
1 / 10
Plan for today
Please note my new oce h
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
October 9, 2014
1/7
Plan for today
Review of last weeks material
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
[email protected]
Fall 2014, Thursdays, 9:0010:50AM
September 4, 2014
1 / 13
Plan for today
Course organization, requ
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 2 (due Oct. 9, 2014)
1. Find the change-of-basis matrix P from the canonical basis S = cfw_e1 , e2 in R2 to the basis
S = cfw_v 1 , v
Math 140 Linear Algebra - Fall 2017 Class Starter 1.1-1.2
Name: < E: N Number:
Date: Section:
You have 10 minutes to complete this class starter: No calculators, electronic devices, notes or other out
Math 140 - Fall 2017 - Homework - Chapter 4
Give complete, well written solutions to the following exercises:
1. (5 points) Construct a nonzero matrix A with the required property or
state why it is i