Page 382 - 35Linear Algebra

382 Movie Scripts

(·iv) Multiplicative associativity. Just as for addition, this is the re-
quirement that the order of bracketing does not matter. We need to
establish whether

x 1 ? x 1
(a.b) · = a · b · .
x 2 x 2
This clearly holds for real numbers a.(b.x) = (a.b).x. The computation is

(a.b) · x 1 = (a.b).x 1 = a.(b.x 1 ) = a. (b.x 1 ) = a · b · x 1 ,
x 2 (a.b).x 2 a.(b.x 2 ) (b.x 2 ) x 2
which is what we want.

(·v) Unity: We need to find a special scalar acts the way we would expect
‘‘1’’ to behave. I.e.

‘‘1’’ · x 1 = x 1 .
x 2 x 2
There is an obvious choice for this special scalar---just the real number
1 itself. Indeed, to be pedantic lets calculate

1 · x 1 = 1.x 1 = x 1 .
x 2 1.x 2 x 2

2
Now we are done---we have really proven the R is a vector space so lets write
a little square to celebrate.

Example of a Vector Space

This video talks about the definition of a vector space. Even though the
defintion looks long, complicated and abstract, it is actually designed to
model a very wide range of real life situations. As an example, consider the
vector space
V = {all possible ways to hit a hockey puck} .
The different ways of hitting a hockey puck can all be considered as vectors.
You can think about adding vectors by having two players hitting the puck at
the same time. This picture shows vectors N and J corresponding to the ways
Nicole Darwitz and Jenny Potter hit a hockey puck, plus the vector obtained
when they hit the puck together.

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