Linear Algebra

116 Chapter 2. Vector Spaces
1.13 Definition In a vector space with basis B the representation of �v with
respect to B is the column vector of the coefficients used to express �v as a linear
combination of the basis vectors. That is,
⎛ ⎞
c1
⎜c2⎟
⎜ ⎟
RepB(�v) = ⎜ .
⎝ .
⎟
. ⎠
where B = 〈 � β1,..., � βn〉 and �v = c1 � β1 + c2 � β2 + ··· + cn � βn. The c’s are the
coordinates of �v with respect to B.
1.14 Example In P3, with respect to the basis B = 〈1, 2x, 2x 2 , 2x 3 〉, the rep-
resentation of x + x 2 is
cn
B
RepB(x + x 2 ⎛ ⎞
0
⎜
)= ⎜1/2
⎟
⎝1/2⎠
0
(note that the coordinates are scalars, not vectors). With respect to a different
basis D = 〈1+x, 1 − x, x + x 2 ,x+ x 3 〉, the representation
is different.
RepD(x + x 2 ⎛ ⎞
0
⎜
)= ⎜0
⎟
⎝1⎠
0
1.15 Remark This use of column notation and the term ‘coordinates’ has
both a down side and an up side.
The down side is that representations look like vectors from R n , and that
can be confusing when the vector space we are working with is R n , especially
since we sometimes omit the subscript base. We must then infer the intent from
the context. For example, the phrase ‘in R2 , where
� �
3
�v = , ... ’
2
refers to the plane vector that, when in canonical position, ends at (3, 2). To
find the coordinates of that vector with respect to the basis
� � � �
1 0
B = 〈 , 〉
1 2
we solve
� � � � � �
1 0 3
c1 + c2 =
1 2 2
D
B