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Inner Products. Orthogonality. Part 1
Mark as completed
Almaz Khusnutdinov
•
Math enthusiast, love coding and mathematics
Published 9/27/2025
Functional Products. Bases
Problem 1.1
Let
V
=
R
2
[
t
]
with
⟨
f
,
g
⟩
=
∫
0
1
f
(
t
)
g
(
t
)
d
t
.
Find a basis of
W
=
{
p
∈
V
:
⟨
p
,
t
2
−
t
⟩
=
0
}
.
Answer:
{
2
t
−
1
,
10
t
2
−
3
}
.
\text{Let }V=\mathbb{R}_2[t]\text{ with }\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt.\ \text{Find a basis of }W=\{p\in V:\ \langle p,\ t^2-t\rangle=0\}.\\[8pt] \textbf{Answer: }\{\,2t-1,\ 10t^2-3\,\}.
Let
V
=
R
2
[
t
]
with
⟨
f
,
g
⟩
=
∫
0
1
f
(
t
)
g
(
t
)
d
t
.
Find a basis of
W
=
{
p
∈
V
:
⟨
p
,
t
2
−
t
⟩
=
0
}
.
Answer:
{
2
t
−
1
,
10
t
2
−
3
}
.
Problem 1.2
Let
V
=
R
2
[
t
]
with
⟨
f
,
g
⟩
=
∫
0
1
f
(
t
)
g
(
t
)
d
t
.
Find a basis of
W
=
{
p
∈
V
:
⟨
p
,
1
⟩
=
⟨
p
,
t
⟩
=
0
}
.
Answer:
{
t
2
−
t
+
1
6
}
.
\text{Let }V=\mathbb{R}_2[t]\text{ with }\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt.\ \text{Find a basis of }W=\{p\in V:\ \langle p,1\rangle=\langle p,t\rangle=0\}.\\[8pt] \textbf{Answer: }\{\,t^2-t+\tfrac16\,\}.
Let
V
=
R
2
[
t
]
with
⟨
f
,
g
⟩
=
∫
0
1
f
(
t
)
g
(
t
)
d
t
.
Find a basis of
W
=
{
p
∈
V
:
⟨
p
,
1
⟩
=
⟨
p
,
t
⟩
=
0
}
.
Answer:
{
t
2
−
t
+
6
1
}
.
Problem 1.3
Let
V
=
R
2
[
t
]
with weighted inner product
⟨
f
,
g
⟩
=
∫
0
1
t
f
(
t
)
g
(
t
)
d
t
.
Find a basis of
W
=
{
p
∈
V
:
⟨
p
,
t
⟩
=
0
}
.
Answer:
{
4
t
−
3
,
5
t
2
−
3
}
.
\text{Let }V=\mathbb{R}_2[t]\text{ with weighted inner product }\langle f,g\rangle=\int_0^1 t\,f(t)g(t)\,dt.\ \text{Find a basis of }W=\{p\in V:\ \langle p,\ t\rangle=0\}.\\[8pt] \textbf{Answer: }\{\,4t-3,\ 5t^2-3\,\}.
Let
V
=
R
2
[
t
]
with weighted inner product
⟨
f
,
g
⟩
=
∫
0
1
t
f
(
t
)
g
(
t
)
d
t
.
Find a basis of
W
=
{
p
∈
V
:
⟨
p
,
t
⟩
=
0
}
.
Answer:
{
4
t
−
3
,
5
t
2
−
3
}
.
Table of Contents
Functional Products. Bases
Problem 1.1
Problem 1.2
Problem 1.3
