Linear Algebra, 2020a

124 Chapter Two. Vector Spaces
terms, so that the two sums combine the same ⃗β’s in the same order: ⃗v =
c 1
⃗β 1 + c 2
⃗β 2 + ···+ c n
⃗β n and ⃗v = d 1
⃗β 1 + d 2
⃗β 2 + ···+ d n
⃗β n .Now
holds if and only if
c 1
⃗β 1 + c 2
⃗β 2 + ···+ c n
⃗β n = d 1
⃗β 1 + d 2
⃗β 2 + ···+ d n
⃗β n
(c 1 − d 1 )⃗β 1 + ···+(c n − d n )⃗β n = ⃗0
holds. So, asserting that each coefficient in the lower equation is zero is the same
thing as asserting that c i = d i for each i, that is, that every vector is expressible
as a linear combination of the ⃗β’s in a unique way.
QED
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:
⎛ ⎞
c 1
c 2...
Rep B (⃗v) =
⎜ ⎟
⎝ ⎠
c n
where B = 〈⃗β 1 ,...,⃗β n 〉 and ⃗v = c 1
⃗β 1 + c 2
⃗β 2 + ···+ c n
⃗β n . The c’s are the
coordinates of ⃗v with respect to B.
1.14 Example In P 3 , with respect to the basis B = 〈1, 2x, 2x 2 ,2x 3 〉, the representation
of x + x 2 is
⎛ ⎞
0
Rep B (x + x 2 1/2
)= ⎜ ⎟
⎝1/2⎠
0
because x + x 2 = 0 · 1 +(1/2) · 2x +(1/2) · 2x 2 + 0 · 2x 3 . With respect to a
different basis D = 〈1 + x, 1 − x, x + x 2 ,x+ x 3 〉, the representation is different.
⎛ ⎞
0
Rep D (x + x 2 0
)= ⎜ ⎟
⎝1⎠
0
1.15 Remark Definition 1.1 requires that a basis be a sequence so that we can
write these coordinates in an order.
When there is only one basis around, we often omit the subscript naming
that basis.
B
D
B