Crdinates 161

Coordinates

Coordinates in linear algebra are a bit different from the coordinates explained

in high school. I’ll try explaining the difference between the two using the image

below.

O

x

2

4

7

x

1

O

u

2

x

2

1

2

u

1

x

1

x

O

point (7, 4)

point

0

1

+ 4

1

0

7

4

= 7

point (2, 1)

1

2

+ 1

3

1

7

4

= 2

When working with coordinates and coordinate systems at the high school

level, it’s much easier to use only the trivial basis:

, … ,

1

0

0

0

1

0

0

0

1

,

In this kind of system, the relationship between the origin and the point in

the top right is interpreted as follows:

O

x

2

4

7

x

1

O

u

2

x

2

1

2

u

1

x

1

x

O

point (7, 4)

point

0

1

+ 4

1

0

7

4

= 7

point (2, 1)

1

2

+ 1

3

1

7

4

= 2

162 Chapter 6 More Vectors

It is important to understand that the trivial basis is only one of many bases

when we move into the realm of linear algebra—and that using other bases pro-

duces other relationships between the origin and a given point. The image below

illustrates the point (2, 1) in a system using the nontrivial basis consisting of

the two vectors

1

2

and u

2

=

3

1

u

1

=

.

This alternative way of thinking about coordinates is very useful in factor

analysis, for example.

O

x

2

4

7

x

1

O

u

2

x

2

1

2

u

1

x

1

x

O

point (7, 4)

point

0

1

+ 4

1

0

7

4

= 7

point (2, 1)

1

2

+ 1

3

1

7

4

= 2

Start Free Trial

No credit card required