Let p 1 ( x ) = 1 + x , p 2 ( x ) = − x + x 2 , and p 3 ( x ) = 1 + 2 x 2 . Show that { p 1 , p 2 , p 3 } is a basis for P 2 ( ℝ ) . [ Hint: You need not show that the set is both linearly independent and a spanning set for P 2 ( ℝ ) . Use a theorem from this section to shorten your work.]
Let p 1 ( x ) = 1 + x , p 2 ( x ) = − x + x 2 , and p 3 ( x ) = 1 + 2 x 2 . Show that { p 1 , p 2 , p 3 } is a basis for P 2 ( ℝ ) . [ Hint: You need not show that the set is both linearly independent and a spanning set for P 2 ( ℝ ) . Use a theorem from this section to shorten your work.]
Solution Summary: The author explains that a set of vectors leftp_1,.p.2, is its basis for
Let
p
1
(
x
)
=
1
+
x
,
p
2
(
x
)
=
−
x
+
x
2
, and
p
3
(
x
)
=
1
+
2
x
2
. Show that
{
p
1
,
p
2
,
p
3
}
is a basis for
P
2
(
ℝ
)
. [Hint: You need not show that the set is both linearly independent and a spanning set for
P
2
(
ℝ
)
. Use a theorem from this section to shorten your work.]
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