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 of IR. What is that number n here?
:c) What is the dim
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 characteristics of a population and
give a geometrical interp
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 hints, then start a discussion on
Piazza or in office hours
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 some i R. Clearly, if ~vi is an eigenvector of A, then
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) [3 points] Find all values of x such that the matrix A (
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 cross product of two vectors in R3 is a useful operation
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 hints, then start a discussion on
Piazza or in office hours
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
~ = a1 v~1 + a2 v~2 + . . . + am v~m .
span(v~1 , v~2 , . .
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 examples; in each case, write down the system of linear equ
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 , without
3
1
4
doing any calculations, where ~x =
and 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 hints, then start a discussion on
Piazza or in office hours
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
mathematical arguments (background handout for courses requirin
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 commutative: ~x ~y = ~y ~x and distributive: (~x + ~y) ~v = ~x
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 v~1 + a2 v~2 + . . . + am v~m .
span(v~1 , v~2 , . . .
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 ~x(0) = ~x0 .
We can write this as a direct formula
~x(
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 that is the closest
possible to ~x, out of all the vec
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 ~x = (u1 ~x)u1 + (u2 ~x)u2 + . . . + (um ~x)um ,
where u
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 hints, then start a discussion on
Piazza or in office hours
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 examples; in each case, write down the system of linear equa
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 eigenvalues.
(c) [2 points] Is A diagonalizable? Justify your ans
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 hints, then start a discussion on
Piazza or in office hours
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 transformations, to use a basis other than the standard b
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
mathematical arguments (background handout for courses requirin
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 hints, then start a discussion on
Piazza or in office hours
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 parts of the questions is given. Please note that the poin
33A/1 Linear Algebra: Homework 4
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 hints, then start a discussion on
Piazza or in office hours
33A/1 Linear Algebra: Notes 5
Puck Rombach
Last updated: November 9, 2015
Change of Basis
We are so used to working in the standard basis that we almost forget it is there. However, it is
often not the best choice of basis for a given linear transformatio
33A/1 Linear Algebra: Practice Midterm Exam 2
Version A
Name:
UID:
1
2
Question 1
(a) [3 points] Explain what it means for two n n matrices A and B to be similar.
(b) [3 points] Show that if A is similar to B, then Ak is similar to Bk , for all integers k