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Lecture

Week 1 (lec 1-3) Lecture Notes

by OneClass31850 , Winter 2012
9 Pages
169 Views

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
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
.
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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Description
Lec 1 Vectors inWednesday January 04 2012934 AM Rather than view as the set of all points with real coordinatesie We will view as a set of vectorsVectors will be written mainly as column matricesNew definition for Linear AlgebraThe origin is also known as the zero vectorWhy vectorsYou can perform operations in with vectors that you cant do with pointsDefinitionLet andbe two vectors inLet We define vector addition as We define scalar multiplicationasEg If and then DefinitionLetIf for then is a linear combinationof Egis a linear combination ofMATH 136 Page 1
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