Problem Set 17 - Finding the Eigenvalues and Eigenvectors of a Matrix
1. (a) For each of the matrices below, find all (real) eigenvalues. Then find a basis of each eigenspace,
and find an eigenbasis i
Problem Set 21 - Introduction to Continuous Dynamical Systems
1. Modeling population growth with differential equations.(1)
(a) The simplest model of population growth is often credited to Thomas Malt
Problem Set 12 - Orthogonal Projections and Orthonormal Bases
1
1
W1. (a) Find the angle between 2 and 1.
1
1
1
1
Solution. Let ~v = 2 and w
~ = 1, and let be the angle between ~v and w.
~ The
Problem Set 13 - The Gram-Schmidt Process, The Transpose of a Matrix
2
4
6
1. Let ~v1 = 2, ~v2 = 0, ~v3 = 2.
1
1
1
(a) Perform the Gram-Schmidt process (by hand) on the given vectors.
Solution. We
Problem Set 24 - Nonlinear Continuous Dynamical Systems
1. In this problem, youll look at the nonlinear system
dx
=y2x
dt
dy = x2 y
dt
(a) Do a qualitative phase plane analysis of the system. That is
Problem Set 22 - Linear Continuous Dynamical Systems and the Matrix
Exponential
1. Being flexible with the different forms of the solutions of dynamical systems is helpful when solving
problems.
(6 po
10/7/2010 FIRST HOURLY PRACTICE IV
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
Solutions
Math 21b, Spring 09
1. The verication that cos(nx), sin(nx), 1/ 2 form an orthonormal family is a straightforward computation, when using the identities provided. For example, cos(nx), sin(m
MATH 21B: UNIT TWO REVIEW
(1) Gram-Schmidt and QR Decomposition (Worksheet 11)
A collection of vectors ~u1 , . . . , ~un is called orthonormal if they are all mutually orthogonal
and have length one.
ODE COOKBOOK
x x = 0
Math 21b, 2017
x(t) = Cet
This first order ODE is by far the most important differential equation. A linear system of
differential equation x (t) = Ax(t) reduces to this after dia
Introduction to Linear Transformations
Many branches of mathematics are concerned with studying functions with particular properties. For
example, single variable calculus is largely concerned with st
The Heat and Wave Equations
1. Suppose f (t, x), defined for x in [0, ], is a function satisfying the boundary conditions f (t, 0) =
f (t, ) = 0.
(a) How can we use Fourier sine series to express f (t
MATH 21B: UNIT ONE REVIEW
(1) Introduction to Systems of Linear Equations
(2) Gauss-Jordan Elimination
(3) On solutions of linear systems
We looked at systems of linear equations A~x = ~b, and how we
Math 21b: Unit Three Review
In this nal unit, we applied the theory of eigenvalues and eigenvectors (from Unit 2) to the study of
dierential equations and continuous dynamical systems.
1. Continuous D
11/4/2010 SECOND HOURLY PRACTICE VI
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
FINAL PRACTICE IV, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page a
FINAL PRACTICE III, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page
Problem Set 4 - How much data do you need to determine a linear
transformation?
Learning Objectives:
You should understand what we mean by the matrix of a linear transformation. (When we say that
you
Problem Set 5 - Warmup Solutions
The first problem below (W1) is a warmup. Weve posted the answers online already so that you can check
that youre comfortable visualizing projections before doing the
Problem Set 9 - Image and Kernel of a Linear Transformation,
Introduction to Linear Independence
Learning Objectives:
You should know and understand the definitions of the image and kernel of a linea
Diagonalization
3
1. You are given the following information about a 3 3 matrix A: ~v1 = 1 is an eigenvector of A
0
1
0
with eigenvalue 3, ~v2 = 0 is an eigenvector of A with eigenvalue 2, and ~
Problem Set 12 - Orthogonal Projections and Orthonormal Bases
Learning Objectives:
You should understand how we use the dot product to define geometric ideas like the length of a
vector and the angle
Problem Set 3 - Introduction to Linear Transformations
Learning Objectives:
You should be able to state the definition of a linear transformation. (Being able to state and interpret
definitions will
Problem Set 6 - More on Bases of Rn , Matrix Products
Learning Objectives:
You should understand that n vectors ~v1 , . . . , ~vn form a basis of Rn rref ~v1 ~vn = In
every vector ~x in Rn can be ex
Linear Combinations and Linear Transformations
As youve probably already noticed, the two fundamental operations were interested in in linear algebra
are addition and scalar multiplication. In particu
Problem Set 8 - Coordinates
Learning Objectives:
If B is a basis of Rn , you should understand what we mean by the B-coordinates of a vector ~x in
Rn , and you should be able to compute [~x]B for a g
Problem Set 7 - Matrix Inverses
Learning Objectives:
You should know the definition of an invertible linear transformation/matrix and understand what
the inverse linear transformation/matrix represen
Problem Set 10 - Subspaces of Rn , Bases and Linear Independence
Learning Objectives:
You should be able to state the definition of a subspace of Rn , and you should have an intuitive picture
of a su
Problem Set 5 - More Examples of Linear Transformations
Learning Objectives:
You should be familiar with several geometric examples of linear transformations (rotations, scalings,
reflections, projec
Coordinates
2
5
0
1. Let ~v1 = 1 , ~v2 = 3 , ~v3 = 1.
3
11
6
2
(a) Show that B = cfw_~v1 , ~v2 , ~v3 forms a basis of R3 , and express ~x = 1 as a linear combination of
5
these basis vectors.
Problem Set 11 - Dimension and the Rank-Nullity Theorem
Learning Objectives:
You should understand the Rank-Nullity Theorem (and the terms rank and nullity).
You should be able to use the Rank-Nulli