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Lecture

Week 1 (lec 1-3) Lecture Notes


Department
Mathematics
Course Code
MATH136
Professor
Patrick Roh

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Rather than view
as the set of all points with
real coordinates.
(ie.

We will view as a set of vectors.
Vectors will be written mainly as column matrices.
New definition for Linear Algebra:

The origin is also known as the zero vector.
You can perform operations in with vectors that you can't do with points.
Definition:
Let
and
be two vectors in .
Let .
We define vector addition as
We define scalar multiplication as
Eg
If
and
then





Definition:
Let
If
for
then
is a linear combination of
.
Eg






is a linear combination of

Lec 1 - Vectors in 
Wednesday, January 04, 2012
9:34 AM
MATH 136 Page 1

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

is a combination of

There are 10 important properties that hold for vectors in
Theorem:
Let
and
(vector addition is defined)
1.
2.
3.
There exists a vector
where
.
4.
For each
, there exists vector
where

.
5.
(scalar multiplication is defined)
6.

7.
8.
9.

10.
Proof of 8:
Also, note that by properties 1 & 6, is closed under linear combinations.
This means that any linear combination of vectors in is still a vector in .
This is an important property we will want when taking groups of vectors from .
Eg
Let

(the vector equation of )
Let
be the set of all scalar multiples of
Geometrically, if we take t the points that are represented by the vectors in we obtain the x-axis on the xy-plane in
.
MATH 136 Page 2

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.
If we change
to become
for some  is still on the same set.
Since
is in our original set , we have that
is a linear combination of
and vice versa.
is closed under linear combinations.
MATH 136 Page 3
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