Page 379 - 35Linear Algebra
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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
                                                3
                   must replace x → (x, y, z) ∈ R ). 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
                                                           2
                                                       2
                   corresponds to the lines x = ±t ⇒ x − t = 0. This should get you started in
                   your search for vectors with zero length.

                   G.4      Vector Spaces

                   Examples of Each Rule

                                   2
                   Lets show that R 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:
                                                           2
                      Before we start, remember that for R we define vector addition and scalar
                   multiplication component-wise.

                                                                                   x 1       y 1
                   (+i) Additive closure: We need to make sure that when we add         and
                                                                                   x 2       y 2
                                                                                         2
                         that we do not get something outside the original vector space R . 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

                                               x 1    y 1     x 1 + x 2   2
                                                   +      :=           ∈ R .
                                               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.

                                                x 1    y 1  ?  y 1    x 1
                                                    +      =       +      .
                                                x 2    y 2     y 2    x 2
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