linear-guest

G.4 Vector Spaces 379
idea is to plot the story of your life on a plane with coordinates (x, t). The
coordinate x encodes where an event happened (for real life situations, we
must replace x → (x, y, z) ∈ R 3 ). The coordinate t says when events happened.
Therefore you can plot your life history as a worldline as shown:
Each point on the worldline corresponds to a place and time of an event in your
life. The slope of the worldline has to do with your speed. Or to be precise,
the inverse slope is your velocity. Einstein realized that the maximum speed
possible was that of light, often called c. In the diagram above c = 1 and
corresponds to the lines x = ±t ⇒ x 2 − t 2 = 0. This should get you started in
your search for vectors with zero length.
G.4 Vector Spaces
Examples of Each Rule
Lets show that R 2 is a vector space. To do this (unless we invent some clever
tricks) we will have to check all parts of the definition. Its worth doing
this once, so here we go:
Before we start, remember that for R 2 we define vector addition and scalar
multiplication component-wise.
( ) ( )
x1 y1
(+i) Additive closure: We need to make sure that when we add and
x 2 y 2
that we do not get something outside the original vector space R 2 . This
just relies on the underlying structure of real numbers whose sums are
again real numbers so, using our component-wise addition law we have
( ) ( ) ( )
x1 y1 x1 + x
+ :=
2
∈ R 2 .
x 2 y 2 y 1 + y 2
(+ii) Additive commutativity: We want to check that when we add any two vectors
we can do so in either order, i.e.
( ) ( ) ( ) ( )
x1 y1 ? y1 x1
+ = + .
x 2 y 2 y 2 x 2
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