Northwestern University
Name:
Math 290-1: Midterm 1
Fall Quarter 2014
Monday, October 20, 2014
Put a check mark next to your section:
Davis (10am)
Alongi
Graham
Canez
Peterson
Davis (12pm)
Instructions:
Question Possible Score
points
1
20
2
20
3
10
4
15
5
Relation between rank and number of solutions.
Based on the form of the reduced echelon form of a matrix, there is a strong relation between the rank of
a matrix and the number of solutions of a system having that matrix as its coefficients. For instance,
. \m
Northwestern University lamaELL)
Student ID:
Math 290-1 Final Exam
Fall Quarter 2013
Wednesday, December 11, 2013
Put a check mark next to your section:
Allen - H Caez Tl
Broderick 1000+ Davis J
H Broderick 12:00 i
Math 290-1: Lecture Notes, Week 2
Northwestern University, Fall 2013
September 30, 2013: Solutions of Linear Systems
Today we continued talking about solving systems of linear equations, and started talking about
vectors and how they provide an alternate
Math 290-1: Lecture Notes, Week 1
Northwestern University, Fall 2013
September 25, 2013: Introduction to Linear Systems
Today I gave a brief introduction to some concepts well be looking at this quarter, such as matrices,
eigenvalues, and eigenvectors. I
Math 290-1: Lecture Notes, Week 6
Northwestern University, Fall 2013
October 28, 2013: Bases and Dimension
Today we continued talking about the notion of a basis of a subspace of Rn , and introduced the
idea of the dimension of a subspace. The dimension o
Math 290-1: Lecture Notes, Week 3
Northwestern University, Fall 2013
October 7, 2013: Linear Transformations
Today we started talking about what are called linear transformations and their relation to
matrices.
Warm-Up. Is the vector
4
0
2
a linear combin
Math 290-1: Lecture Notes, Week 5
Northwestern University, Fall 2013
October 21, 2013: Midterm 1
October 23, 2013: Subspaces of Rn
Today we started talking about the notion of a subspace of Rn . The denition is kind of abstract,
but in the end subspaces a
Math 290-1: Lecture Notes, Week 4
Northwestern University, Fall 2013
October 14, 2013: Invertibility and Inverses
Today we spoke about the notion of a matrix being invertible, and nding the inverse of a matrix
which is invertible. We will continue looking
:3: a A; . , Name:
,9. Northwestern Univerer Student n)-
Math 290-1 Final Exam Practice
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) Suppose that T is an invertible linear transformation from R" to R
Math 290-1: Lecture Notes, Week 7
Northwestern University, Fall 2013
November 4, 2013: Determinants
Today we started talking about determinants, which we will continue with all this week. For now all
we are interested in is computing determinants; we will
Lecture 10 (20150420): Change of Coordinates (part 2)
(5.5)
Today:
Cylindrical and Spherical Coordinates
Example 41 (ASN). If f : R2 R is a continuous function, then
4
0
3
0
f (r cos(), r sin()rdrd =
f (x, y)dA.
x2 +y 2 9
This is sometimes true. For an ex
Northwestern University
Name:
Math 290-3 Midterm 1 Practice Problems
Spring Quarter 2015
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) For a continuous function f (x, y),
f (x,0)+1
b
dy dx = b a.
a
f (x,0
Lecture 9 (20150417): Change of Coordinates (part 1)
(5.5)
Today:
Coordinate Changes in R2 and R3
Example 35. Compute the area of the cardioid in R2 which is bounded by the curve
x2 + y 2 = x2 + y 2 + x.
We can convert this curve to polar coordinates by s
Math 290-1: Lecture Notes
Northwestern University, Fall 2013
Written by Santiago Caez
n
These are lecture notes for Math 290-1, the rst quarter of MENU: Linear Algebra and Multivariable Calculus, taught at Northwestern University in the fall of 2013. The
MATH 290 Homework 4
Angelo Lee
1. 2.3.14. For the matrices
1 1
,B = 1 2 3 ,
1 1
1 0 1
1
C = 2 1 0 , D = 1 , E = 5 ,
3 2 1
1
A=
determine which of the 25 matrix products AA, AB, AC, , ED, EE are dened, and compute those that are dened.
Answer: Let M be a
Math 290 - Practice Problems for Test 1
UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT.
3 4
5
5 2 and b = 1. Show that b Spancfw_c1 , c2 by
1. Let c1 and c2 be the columns of A =
6 6
6
writing b as a linear combination of c1 and c2 .
2. Find the value
Northwestern University
Name:
Math 290-1 Midterm 1 Practice Problems
Fall Quarter 2014
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) If A is a 5 3 matrix with rank 3, then Ax = 0 has a unique solution.
(b
Northwestern University
Name:
Student ID:
Math 290-1 Midterm 1 Practice Problems
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) There is a matrix A such that the transformation T from R3 to R2 given by
Northwestern University
Name:
Math 290-1 Practice Midterm 1
Fall Quarter 2015
Monday, October 19, 2015
Note: The actual midterm has 6 true/false questions and 5 computation questions. There are
more problems than that on this practice exam, to give you ex
Lecture 7 (20150413): Triple Integrals (part 3, 5.4)
Today:
Changing the Order of Integration
Example 26 (ASN). For k 0 an integer,
3
3
9y 2
9y 2
9z 2 y 2
0
(y 2 + 1)y k dxdzdy < 0.
This is never true. Indeed, this region is the half of the (solid) sphere
Lecture 8 (20150415): Integration in Polar Coordinates
(5.5)
Today:
Integration in Polar Coordinates
2
Example 29 (SA/C). Write the integral I =
dydzdx.
0
y2
2
x
0
f (x, y, z)dzdydx in the order
The region of integration of this integral is the same as th
Northwestern University
Name:
Student ID:
Math 290-3 Final Exam Solutions
Spring Quarter 2014
Tuesday, June 10, 2014
Math 290-3 Final Exam Solutions
Spring Quarter 2014
Page 2 of 11
1. Determine whether each of the following statements is TRUE or FALSE. J
2. Determine whether each of the following statements is ALWAYS true, SOMETIMES true, or NEVER
true. Justify your answer.
Z
(a). For a continuous function f : R2 ! R,
3
3
Z
x2
yf (x2 , y 2 ), dydx = 0.
0
This is sometimes true.
For an example where it is
Math 290-1: Lecture Notes, Week 8
Northwestern University, Fall 2013
November 11, 2013: Eigenvalues
Today we started talking about eigenvalues of eigenvectors, which make up our last topic for
the quarter, and probably the most important as well. Eigenval
Math 290-1: Lecture Notes, Week 9
Northwestern University, Fall 2013
November 18, 2013: Midterm 2
November 20, 2013: Diagonalization
Today we started talking about what it means for a matrix to be diagonalizable, which weve
actually been secretly talking
Math 290-1: Lecture Notes, Week 10
Northwestern University, Fall 2013
November 25, 2013: Complex Eigenvalues
Today we spoke about complex eigenvalues and eigenvectors. The point is that this all works the
same way as real eigenvalues and eigenvectors do,
Northwestern University
Name:
Math 290-1 Practice Midterm 1
Fall Quarter 2015
Monday, October 19, 2015
The following problems are True or False. Write the word TRUE or the word FALSE. If
the statement is true, prove it. If the statement is false, provide
(7) Let
0 1
1
@
~v1 = 1A
1
and
0
1
2
~v2 = @ 3 A .
1
Find a vector ~v3 in R3 such that ~v1 , ~v2 , ~v3 are linearly independent, or explain why it is
not possible to do so.
Since we know that the dimension of R3 is 3, any basis of R3 must have three linea
3/ 15"];ij
, L Northwestern University
Name:
Math 290-1 Midterm 2
Fall Quarter 2015
Monday, November 16, 2015
Put a check mark next to your section:
l_-|-
l-l-
I_-|-
Instructions:
0 Read each problem carefully.
0 Write legibly.
0 Show all your