Page 14 - 35Linear Algebra

14 What is Linear Algebra?

guess how to multiply a vector by a scalar. For example

   
1
1
3
1 =   .
 1 
1  
3 3
0 0
A very special vector can be produced from any vector of any kind by
scalar multiplying any vector by the number 0. This is called the zero vector
and is usually denoted simply 0. This gives five very different kinds of zero
from the 5 different kinds of vectors in examples A-E above.
(A) 0(3) = 0 (The zero number)

   
1 0
1
(B) 0   =   (The zero 3-vector)
0
0 0
2
3
(C) 0 (1 + x − 2x + 3x ) = 0 (The zero polynomial)
1
1
3
2

3
2
(D) 0 1 + x− x + x + · · · = 0+0x+0x +0x +· · · (The zero power series)
2! 3!
x
(E) 0 (e ) = 0 (The zero function)
In any given situation that you plan to describe using vectors, you need
to decide on a way to add and scalar multiply vectors. In summary:
Vectors are things you can add and scalar multiply.

Examples of kinds of vectors:

• numbers

• n-vectors

• 2nd order polynomials

• polynomials

• power series

• functions with a certain domain

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