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Unformatted text preview: Proof. By deﬁnition, the set B spans V . This means exactly that every element in V can
be written as a linear combination of the elements of B , that is, we can write an arbitrary
element w 2 V as
m
X
w=
ai v i .
1 We have to show that this expression for w is unique. Assume that we have another
expression for w, say
m
X
w=
bi vi .
1 Subtracting we obtain w w= m
X ( ai bi ) vi = 0. 1 This is a linear relation among the elements of B . The only such relation is the trivial one.
Hence ai = bi . This says that there is only one expression for w as a linear combination
of elements of B . Coordinates
Deﬁnition 1. Let A = {v1 , v2 , · · · vn } be a basis of a subspace of Rm . Let v 2 V , so we
can uniquely write
n
X
v=
ai v i .
1 We say that the coordinates of v are a1 , a2 , · · · , an . We write
01
a1
B a2 C
v=B C .
@ a3 A
·A
Example 2. Let 01
01
✓◆
0
0
1
@1A , e3 = @0A}.
E = {e1 =
e=
02
0
1
01
x
3
@y A we have
This is a basis of R and for an arbitrary element v =
z
01
x
@y A .
v = xe1 + xe2 + xe3 , and so v =
zE
This says that the coordinates of v wrt the basis E are x, y, x.
2 ...
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 Fall '08
 MARKMAN
 Linear Algebra, Algebra

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