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.

Why vectors?

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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