Suppose that { u 1 , u 2 , ⋅ ⋅ ⋅ , u p } is a basis for a subspace W , and suppose that x is in W with x = a 1 u 1 + a 2 u 2 + ⋅ ⋅ ⋅ + a p u p . Show that his representation for x in terms of the basis is unique—that is, if x = b 1 u 1 + b 2 u 2 + ⋅ ⋅ ⋅ + b p u p then b 1 = a 1 , b 2 = a 2 , ... , b p = a p .
Suppose that { u 1 , u 2 , ⋅ ⋅ ⋅ , u p } is a basis for a subspace W , and suppose that x is in W with x = a 1 u 1 + a 2 u 2 + ⋅ ⋅ ⋅ + a p u p . Show that his representation for x in terms of the basis is unique—that is, if x = b 1 u 1 + b 2 u 2 + ⋅ ⋅ ⋅ + b p u p then b 1 = a 1 , b 2 = a 2 , ... , b p = a p .
Solution Summary: The author explains that x is in W with two representations in terms of the basis and uses the fact that basis is a linearly independent set.
Suppose that
{
u
1
,
u
2
,
⋅
⋅
⋅
,
u
p
}
is a basis for a subspace
W
, and suppose that
x
is in
W
with
x
=
a
1
u
1
+
a
2
u
2
+
⋅
⋅
⋅
+
a
p
u
p
. Show that his representation for
x
in terms of the basis is unique—that is, if
x
=
b
1
u
1
+
b
2
u
2
+
⋅
⋅
⋅
+
b
p
u
p
then
b
1
=
a
1
,
b
2
=
a
2
,
...
,
b
p
=
a
p
.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.