Page 122 - 35Linear Algebra

122 Matrices

Given a particular vector and a basis, your job is to write that vector as a sum of
multiples of basis elements. Here an arbitrary vector v ∈ V is just a matrix, so we
write

a b a 0 0 b 0 0 0 0
v = = + + +
c d 0 0 0 0 c 0 0 d

1 0 0 1 0 0 0 0
= a + b + c + d
0 0 0 0 1 0 0 1
2
2
1
1
= a e + b e + c e + d e .
1
1
2
2
1
2
2
1
The coefficients (a, b, c, d) of the basis vectors (e , e , e , e ) encode the information
2
1
2
1
of which matrix the vector v is. We store them in column vector by writing
   
a a
b
b  
2
2
 
1
2
1
1
1
v = a e + b e + c e + d e =: (e , e , e , e ) c =:   .
2
1
1
2  
1
2
1
2
c
 
2  
d d
B
 
a

b a b
 
4
The 4-vector   ∈ R encodes the vector ∈ V but is NOT equal to it!
c c d
 
d
(After all, v is a matrix so could not equal a column vector.) Both notations on the
right hand side of the above equation really stand for the vector obtained by multiplying
the coefficients stored in the column vector by the corresponding basis element and
then summing over them.
Next, lets consider a tautological example showing how to label column
vectors in terms of column vectors:
2
Example 75 (Standard Basis of R )
The vectors

1 0
e 1 = , e 2 =
0 1
2 {1,2}
are called the standard basis vectors of R = R . Their description as functions
of {1, 2} are

1 if k = 1 0 if k = 1
e 1 (k) = , e 2 (k) =
0 if k = 2 1 if k = 2 .
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