LT Week 6 Homework, Lecture 29 Material
(a) Define the graph of a function f : R2 R.
[1 mark]
Consider the function f : R2 R given by
f (x1 , x2 ) = x21 x22 .
(b) On a single copy of the x1 x2 -plane,
LT Week 6 In-class Practice, Lecture 30 Material
Consider a function h : R2 R and a point (a, b, h(a, b) on the graph of h in R3 .
(a) Define the directional derivative hu (a, b) in terms of the gradi
As discussed in the orientation lecture, the role of the lecture slides produced during
lectures is to complement your lecture notes; they are not a substitute for them.
During lectures, I will be mak
Exercises and Solutions from Lecture 3
Exercise 3.7.1 Consider the function f : R ! R defined by
f (x) =
8
< 1
:
x3
(x
3)2 + 1
x>3
(i) Sketch the graph of f on the Cartesian plane.
For the next part,
MATH 304
Linear Algebra
Lecture 14:
Basis and coordinates.
Change of basis.
Linear transformations.
Basis and dimension
Definition. Let V be a vector space. A linearly
independent spanning set for V i
What is a Vector Space?
Georey Scott
These are informal notes designed to motivate the abstract definition of a vector space to
my MAT185 students. I had trouble understanding abstract vector spaces w
Chapter 1
Sets and Functions
We understand a set to be any collection M of certain distinct objects
of our thought or intuition (called the elements of M ) into a whole.
(Georg Cantor, 1895)
In mathem
3. INNER PRODUCT SPACES
3.1. Definition
So far we have studied abstract vector spaces. These are a generalisation of the geometric
spaces 2 and 3. But these have more structure than just that of a vec
Subspaces, basis, dimension, and rank
Math 40, Introduction to Linear Algebra
Wednesday, February 8, 2012
Subspaces of Rn
Subspaces of Rn
OneOne
motivation
fornotion
notion
motivation for
subspaces of
Chapter 1
Sets and Functions
Sets
Mathematicians try very hard to precisely define new concepts using only previously defined concepts. There is, at the beginning of this process, a concept
that is no
LT Week 2 In-class Practice, Lecture 22 Material
Consider the linear system Ax = b where
0
1
1 3 1 0
A = @2 6 0 2 A ,
4 12 1 3
0 1
x1
Bx 2 C
C
x=B
@x 3 A
x4
and
0 1
2
b = @ 6 A.
11
(a) Find the reduce
CS131
Part IV, Calculus
CS131 Mathematics for Computer Scientists II
Note 23
DIFFERENTIATION OF INVERSE FUNCTIONS
Range,injection,surjection,bijection.Letf : A ! B be a function
from a set A to a set
Lecture 2
The rank of a matrix
Eivind Eriksen
BI Norwegian School of Management
Department of Economics
September 3, 2010
Eivind Eriksen (BI Dept of Economics)
Lecture 2 The rank of a matrix
September
MA212: Further Mathematical Methods (Linear Algebra) 201718
Exercises 5: Complex matrices
For these and all other exercises on this course, you must show all your working.
1.
(a) Which, if any, of the
Exercises and Solutions from Lecture 34
Exercise 34.6.1
compounded
(a) Find the effective annual rate of interest on 1000 at 8%
(i) annually
(ii) quarterly
(iii) continously.
(b) Determine the interes
Exercises and Solutions from Lecture 29
Exercise 29.5.1 Using the simpler notation x1 = x, x2 = y, show that the function
f : R2 R given by
f (x, y) =
x3 + y 3
x+y
is homogeneous of degree n = 2. Henc
Exercises and Solutions from Lecture 37
Exercise 37.4.1 Write down a linear differential equation P (D)y = 0 whose
solutions include the function e2x and another such equation whose solution include
t
Exercises and Solutions from Lecture 35
3 i and w = 1 + i as points on the complex
z6
plane. Express z and w in polar exponential form and find q = 10 in both polar
w
exponential and Cartesian form.
E
2
Probability Theory on Coin Toss Space
2.1 Finite Probability Spaces
A nite probability space is used to model a situation in which a random
experiment with nitely many possible outcomes is conducted
Lent 2018 examination
MA100
Lent Term Mock Exam
2017/18, 2016/17 and 2015/16 syllabus
Instructions to candidates
This paper contains 2 questions. Answer both questions. The questions carry equal numbe
MA212: Further Mathematical Methods (Linear Algebra) 201718
Exercises 3: Similar matrices and real inner products
For these and all other exercises on this course, you must show all your working.
1. S
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 7
1. (a) The integral
Z
0
M
1
x
dx = log(1 + M 2 ),
2
1+x
2
tends to infinity as M , which means that the integral
Z
x
dx,
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 6
1. How do you sketch a graph? Do you plot the values of the function at 10 points and join the
dots? Is that what you lea
22 Vector Spaces associated with matrices, 2 of 2
We establish an orthogonality relationship between the null space, N(A) Q R", and
the row space, RS(A) Q R, of an m x n matrix A. Then, we introduce t
21 Vector Spaces associated with matrices, 1 of 2
In addition to the null space N(A) g R" of an m x n matrix A, we introduce its
column space CS(A) Q Rm and its raw space RS(A) Q R and establish a bas
MA212: Further Mathematical Methods (Calculus) 201718
Solutions to Exercises 9
1. For (a), taking the Laplace transforms of both sides of the differential equation, we get
s2 y(s) sy(0) y 0 (0) y(s) =
MA212: Further Mathematical Methods (Calculus) 201718
Solutions to Exercises 7
1. (a) The integral
Z
0
M
1
x
dx = log(1 + M 2 ),
2
1+x
2
tends to infinity as M , which means that the integral
Z
x
dx,
MA212: Further Mathematical Methods (Calculus) 201718
Solutions to Exercises 8
1. What you need is Leibnizs rule for differentiating an integral with respect to a parameter (see
Section 9.6 of the Sub