Page 13 - 35Linear Algebra

1.2 What are Vectors? 13

3
3
2
4
2
(C) Polynomials: If p(x) = 1 + x − 2x + 3x and q(x) = x + 3x − 3x + x then
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their sum p(x) + q(x) is the new polynomial 1 + 2x + x + x .
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1
1
1
2
2
3
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(D) Power series: If f(x) = 1+x+ x + x +· · · and g(x) = 1−x+ x − x +· · ·
2! 3! 2! 3!
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2
then f(x) + g(x) = 1 + 1 x + 1 x · · · is also a power series.
2! 4!
x
(E) Functions: If f(x) = e and g(x) = e −x then their sum f(x) + g(x) is the new
function 2 cosh x.
There are clearly different kinds of vectors. Stacks of numbers are not the
only things that are vectors, as examples C, D, and E show. Vectors of
different kinds can not be added; What possible meaning could the following
have?

9
+ e x
3
In fact, you should think of all five kinds of vectors above as different
kinds, and that you should not add vectors that are not of the same kind.
On the other hand, any two things of the same kind “can be added”. This is
the reason you should now start thinking of all the above objects as vectors!
In Chapter 5 we will give the precise rules that vector addition must obey.
In the above examples, however, notice that the vector addition rule stems
from the rules for adding numbers.
When adding the same vector over and over, for example

x + x , x + x + x , x + x + x + x , . . . ,

we will write
2x , 3x , 4x , . . . ,

respectively. For example

           
1 1 1 1 1 4
4
4   =   +   +   +   =   .
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1
1
1
1
0 0 0 0 0 0
Defining 4x = x + x + x + x is fine for integer multiples, but does not help us
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make sense of x. For the different types of vectors above, you can probably
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