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MATH 33A Midterm 1 Oct. 23, 2017
STUDENT NAME: 7' o N 5 i
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STUDENT ID NUMBER:
DISCUSSION SECTION NUMBER:
E
Directions
Answer each question in the space provided. Please write clearly and legi
Math 33A
PRACTICE FINAL
Problem 1.
(1) True . Indeed, since the columns of Q are orthonormal, we have Q T Q = I. It
follows that:
A T A = R T Q T QR = RIR = R2
(2)
(3)
(4)
(5)
So det( A T A) = det( R2
Midterm (B)
N
July 13 2017
This test totals 40 points and you get 50 minutes to do it. Answer the
questions in the spaces provided on the question sheets. Show work unless
the question says otherwise.
\ a Q t zet
t WW _ Mm
Mmm AWN Com? iLgtiQe 1-99 + 2 7 5 i
4. (10 pts) Find the general solution of the differential equation
:5 Fovcww i m
J
2y + y + 3y =(5t,;
3figgzmmw 4; Assoc. Home
i314 : x
Math 33A
Midterm 2
2010 February 22 - i ' - o 4 . A
Problem 1- Consider the matrix A = 3 6 3 ~21 9 _ Z X
2 3 5 7 1 , if
:8.) Find a. basis of the iInage of A. F
b) The kernel of A is a. subspace
1
33A/1 Linear Algebra and Applications: Practice Final Exam
Name:
UID:
33A/1 Linear Algebra - Puck Rombach
Final Exam
2
Question 1
(a) [4 points] Define the correlation coefficient between two charac
33A/1 Linear Algebra: Homework 7
Puck Rombach
These are solution outlines. They should be sufficient to guide you to a full solution. If
you are unsure how to construct a full solution with these hint
33A/1 Linear Algebra: Notes 6
Puck Rombach
Last updated: November 10, 2015
Eigenvectors and eigenvalues
Let A be an n n matrix. A nonzero vector ~vi Rn is called an eigenvector of A if A~vi = i~vi
for
33A/1 Linear Algebra: Practice Midterm Exam 2
Version B
Name:
UID:
1
2
Question 1
(a) [2 points] Explain how you can use the determinant of a matrix to find out whether the matrix
is invertible.
(b) [
33A/1 Linear Algebra: Notes 7
Puck Rombach
Last updated: November 19, 2015
The cross product
First of all, we use the notation v to mean the unit vector in the direction of ~v. So,
v =
~v
.
k~vk
The c
33A/1 Linear Algebra: Homework 1
Puck Rombach
These are solution outlines. They should be sufficient to guide you to a full solution. If
you are unsure how to construct a full solution with these hint
33A/1 Linear Algebra: Notes 2
Puck Rombach
Last updated: October 28, 2015
Vector Spaces
We define the span fo a set of vectors v~1 , v~2 , . . . , v~m as
~ | a1 , a2 , . . . , am R such that w
~ = a
33A/1 Linear Algebra: Notes 1
Puck Rombach
Last updated: October 28, 2015
Systems of linear equations can represent many things. You have probably come across such a
problem before. Here are some exam
33A/1 Linear Algebra: Homework 5
Puck Rombach
Due: No hand-in; prep for Quiz 1 on 3/5 October
Problem 1
Play around with the demonstration project
at goo.gl/bLkej0.
!
!
! Then use it to find ~x B ,
33A/1 Linear Algebra: Homework 3
Puck Rombach
These are solution outlines. They should be sufficient to guide you to a full solution. If
you are unsure how to construct a full solution with these hint
33A/1 Linear Algebra: Notes 3
Puck Rombach
Last updated: October 28, 2015
Mathematical reasoning
To help you practice with mathematical logic, I have copied a handy table from Introduction to
mathemat
33A/1 Linear Algebra: Notes 4
Puck Rombach
Last updated: November 9, 2015
Dot products
The dot product of two vectors in Rn is defined as
~x ~y = x1 y1 + x2 y2 + . . . xn yn .
Dot products are commuta
33A/1 Linear Algebra: Notes 2
Puck Rombach
Last updated: October 8, 2015
Vector Spaces
We define the span fo a set of vectors v~1 , v~2 , . . . , v~m as
~ | a1 , a2 , . . . , am R such that w
~ = a1
33A/1 Linear Algebra: Notes 10
Puck Rombach
Last updated: December 5, 2015
Discrete Dynamical Systems
A (linear) discrete dynamical system takes the form
~x(t) = A~x(t 1), with some initial condition
33A/1 Linear Algebra: Notes 9
Puck Rombach
Last updated: November 26, 2015
Orthogonal Projections as Approximations
The projection of a vector ~x onto a subspcae V can be thought of as the vector in V
33A/1 Linear Algebra: Notes 8
Puck Rombach
Last updated: November 25, 2015
Gram-Schmidt Algorithm
Previously, we found a formula for projections that does not involve a matrix multiplication:
pro jV ~
33A/1 Linear Algebra: Homework 8
Puck Rombach
These are solution outlines. They should be sufficient to guide you to a full solution. If
you are unsure how to construct a full solution with these hint
33A/1 Linear Algebra: Notes 1
Puck Rombach
Last updated: October 5, 2015
Systems of linear equations can represent many things. You have probably come across such a
problem before. Here are some examp
1
Question 1
0 0 0
Let A = 1 1 0 .
1 1 1
(a) [2 points] Find all eigenvalues of A and their algebraic multiplicities.
(b) [3 points] Find the eigenspaces and geometric multiplicities of the eigenvalue
33A/1 Linear Algebra: Homework 9
Puck Rombach
These are solution outlines. They should be sufficient to guide you to a full solution. If
you are unsure how to construct a full solution with these hint
33A/1 Linear Algebra: Practice Midterm Exam 2
Version C
Name:
UID:
1
2
Question 1
(a) [3 points] Explain why it may sometimes be efficient in terms of computational resources, when
computing linear tr
33A/1 Linear Algebra: Notes 3
Puck Rombach
Last updated: October 10, 2015
Mathematical reasoning
To help you practice with mathematical logic, I have copied a handy table from Introduction to
mathemat
33A/1 Linear Algebra: Homework 6
Puck Rombach
These are solution outlines. They should be sufficient to guide you to a full solution. If
you are unsure how to construct a full solution with these hint
33A/1 Linear Algebra: Midterm Exam 1 A
PRACTICE EXAM: ANSWERS
Name:
UID:
This exam has 3 questions. They are each worth 10 points for a total of 30 points. The distribution of points for different par