Page 125 - 35Linear Algebra

7.1 Linear Transformations and Matrices 125

which is actually a simpler column vector! The fact that there are many
bases for any given vector space allows us to choose a basis in which our
computation is easiest. In any case, the standard basis only makes sense
n
for R . Suppose your vector space was the set of solutions to a differential
equation–what would a standard basis then be?

Example 77 (A Basis For a Hyperplane)
Lets again consider the hyperplane

     
1 0
 

V = c 1   + c 2   c 1 , c 2 ∈ R
1
1

0 1
 
One possible choice of ordered basis is
   
1 0
b 1 =   , b 2 =   , B = (b 1 , b 2 ).
1
1
0 1
With this choice
     
1 0 x

x
1
1
:= xb 1 + yb 2 = x   + y   =  x + y  .
y
B 0 1 y
E
0
With the other choice of order B = (b 2 , b 1 )
     
0 1 y

x
1
1
:= xb 2 + yb 1 = x   + y   =  x + y  .
y
B 0 1 0 x
E
We see that the order of basis elements matters.
Finding the column vector of a given vector in a given basis usually
amounts to a linear systems problem:
Example 78 (Pauli Matrices)
Let

z u
V = z, u, v ∈ C
v −z
be the vector space of trace-free complex-valued matrices (over C) with basis

B = (σ x , σ y , σ z ) ,

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