Page 245 - Matrix Analysis & Applied Linear Algebra

240 Chapter 4 Vector Spaces

It would be wrong to infer from Example 4.7.1 that all linear transformations
can be represented by matrices (of finite size). For example, the differential and
integral operators do not have matrix representations because they are defined
on infinite-dimensional spaces. But linear transformations on finite-dimensional
spaces will always have matrix representations. To see why, the concept of “co-
ordinates” in higher dimensions must first be understood.
Recall that if B = {u 1 , u 2 ,..., u n } is a basis for a vector space U, then
each v ∈U can be written as v = α 1 u 1 + α 2 u 2 + ··· + α n u n . The α i ’s in this
expansion are uniquely determined by v because if v = α i u i = β i u i ,
i i
then 0 = (α i − β i )u i , and this implies α i − β i = 0 (i.e., α i = β i ) for each
i
i because B is an independent set.

Coordinates of a Vector
Let B = {u 1 , u 2 ,..., u n } be a basis for a vector space U, and let v ∈U.
The coefficients α i in the expansion v = α 1 u 1 +α 2 u 2 +···+α n u n are
called the coordinates of v with respect to B, and, from now on,
[v] B will denote the column vector

 
α 1
 α 2 
[v] B =  .  .
.
 . 
α n

Caution! Order is important. If B is a permutation of B, then [v] B
is the corresponding permutation of [v] B .

From now on, S = {e 1 , e 2 ,..., e n } will denote the standard basis of unit
n n
vectors (in natural order) for (or C ). If no other basis is explicitly men-
tioned, then the standard basis is assumed. For example, if no basis is mentioned,
and if we write
 
8
7
v =   ,
4
then it is understood that this is the representation with respect to S in the
sense that v =[v] S =8e 1 +7e 2 +4e 3 . The standard coordinates of a vector
are its coordinates with respect to S. So, 8, 7, and 4 are the standard coordinates
of v in the above example.

Example 4.7.2
3
Problem: If v is a vector in whose standard coordinates are