Meaning of matrix multiplication
In these examples we will explore the eect of matrix multiplication on the xy-plane.
4 1
Example 1: The matrix A =
transforms the unit square into a parallelogram
1 3
as follows.
The unit square has sides i and j. In order
Determinants 1.
Given a square array A of numbers, we associate with it a number called the determinant
of A, and written either det(A), or |A|. For 2 2
a
c
(1)
b
d
= ad bc.
Do not memorize this as a formula learn instead the pattern which gives the terms
Matrices 3. Homogeneous and Inhomogeneous Systems
Theorems about homogeneous and inhomogeneous systems.
On the basis of our work so far, we can formulate a few general results about square
systems of linear equations. They are the theorems most frequently
Components and Projection
b
b
If A is any vector and u is a unit vector then the component of A in the direction of u is
b
A u.
(Note: the component is a scalar.)
b
If is the angle between A and u then since |b | = 1
u
b
A u = |A|b | cos = |A| cos .
u
The
Keplers Second Law
By studying the Danish astronomer Tycho Brahes data about the motion of the planets,
Kepler formulated three empirical laws; two of them can be stated as follows:
Second Law A planet moves in a plane, and the radius vector (from the sun
Matrices 2. Solving Square Systems
of Linear Equations; Inverse Matrices
Solving square systems of linear equations; inverse matrices.
Linear algebra is essentially about solving systems of linear equations, an important
application of mathematics to real
Determinants 2. Area and Volume
B
Area and volume interpretation of the determinant:
(1)
a
1
b1
a2
b2
= area of parallelogram with edges A = (a1 , a2 ), B = (b1 , b2 ).
C
(2)
a1
b1
c1
a2
b2
c2
A
a3
b3
c3
B
A
= volume of parallelepiped with edges row-vec
8.1
A familx of non-analytic functions.
Let m 2 0 be any nonnegative integer. Define
e-l/X2
-*~E for x # 0,
x
fm(X) =
0 for x = 0.
We will show that each of the functions fm(x) has continuous
derivatives of-all orders, for all x. We also show that none
of
18.02 Exam 1
Problem 1.
Let P , Q and R be the points at 1 on the x-axis, 2 on the y-axis and 3 on the z-axis, respectively.
!
!
a) (6) Express QP and QR in terms of j and k.
i,
b) (9) Find the cosine of the angle P QR.
Problem 2. Let P = (1, 1, 1), Q =
18 014 Problem Set 5 Solutions
Total: 24 points
Problem : Let f (x) = x4 + 2x2 + 1 for 0 x 10.
(a) Show f is strictly increasing; what is the domain of its inverse function g?
(b) Find an expression for g, using radicals.
Solution (4 points) (a) Let 0 x <
Matrices 1. Matrix Algebra
Matrix algebra.
Previously we calculated the determinants of square arrays of numbers. Such arrays are
important in mathematics and its applications; they are called matrices. In general, they
need not be square, only rectangula
18.02 Exam 1 Solutions
Problem 1.
!
a) P = (1, 0, 0), Q = (0, 2, 0) and R = (0, 0, 3). Therefore QP =
! !
QP QR
h1, 2, 0i h0, 2, 3i
4
p
b) cos = ! ! = p
=p
2 + 22 22 + 32
65
1
QP QP
!
2 and QR =
2 + 3k.
Problem 2.
!
!
a) P Q = h 1, 2, 0i, P R = h 1, 0, 3
R.l
The basic theorems on power series.
Whenever we have a series 2 un(x) of functions, there
are three fundamental questions we ask:
(1) Given the series 2 un(x), for what values of x
does the series converge?
(2) Given 2 un(x), if it converges to a func
L'Hooital's rule for 0/0
Theorem. Suppose f(x) -> 0 and g(x) > O as
x>a- g
'_h
N
Q.
E
I
then also f(x)/g(x) > L as x > a.
This result holds whether a and -L are finite or
infinite, and it also holds if the limits are one-sided.
_._._
Proof. The
Cross Product
The cross product is another way of multiplying two vectors. (The name comes from the
symbol used to indicate the product.) Because the result of this multiplication is another
vector it is also called the vector product.
As usual, there is
Vectors
Our very rst topic is unusual in that we will start with a brief written presentation. More
typically we will begin each topic with a videotaped lecture by Professor Auroux and follow
that with a brief written presentation.
As we pointed out in th
18.02 Practice Exam 1
z
Problem 1. (15 points)
A unit cube lies in the rst octant, with a vertex at the origin (see gure).
!
!
a) Express the vectors OQ (a diagonal of the cube) and OR
(joining O to the center of a face) in terms of k.
, ,
x
b) Find the
Dot Product
The dot product is one way of combining (multiplying) two vectors. The output is a
scalar (a number). It is called the dot product because the symbol used is a dot. Because
the dot product results in a scalar it, is also called the scalar prod
Equations of planes
We have touched on equations of planes previously. Here we will ll in some of the details.
Planes in point-normal form
The basic data which determines a plane is a point P0 in the plane and a vector N orthogonal
to the plane. We call N
18.02 Practice Exam 1 Solutions
Problem 1.
1
!
! 1
a) OQ = + + k; OR = + + k.
2
2
p
! !
h1, 1, 1i h 1 , 1, 1 i
2 2
OQ OR
2
2
b) cos = ! ! =
=
.
p q3
3
|OQ | |OR |
3 2
Problem 2.
~
dR
~
Velocity: V =
= h 3 sin t, 3 cos t, 1i.
dt
Problem 3.
2
6
a) Minor
Q.l
0 Notes on error estimates.
Each of the standard convergence tools for series brings
with it a method for estimating the error made in approximating
the limit by taking only finitely many terms of the series. We
treat this method for both the integral