Problems Let { u 1 , u 2 , u 3 } be linearly independent vectors in an inner product space V , and suppose that u 1 and u 2 are orthogonal. Define the vector u 3 in V by u 3 = v + λ u 1 + μ u 2 Where λ , μ are scalars. Derive the values of λ and μ such that { u 1 , u 2 , u 3 } is an orthogonal basis for the subspace of V spanned by { u 1 , u 2 , v } .
Problems Let { u 1 , u 2 , u 3 } be linearly independent vectors in an inner product space V , and suppose that u 1 and u 2 are orthogonal. Define the vector u 3 in V by u 3 = v + λ u 1 + μ u 2 Where λ , μ are scalars. Derive the values of λ and μ such that { u 1 , u 2 , u 3 } is an orthogonal basis for the subspace of V spanned by { u 1 , u 2 , v } .
Solution Summary: The author explains that the values of lambda and mu are orthogonal if they are scalars.
Let
{
u
1
,
u
2
,
u
3
}
be linearly independent vectors in an inner product space
V
, and suppose that
u
1
and
u
2
are orthogonal. Define the vector
u
3
in
V
by
u
3
=
v
+
λ
u
1
+
μ
u
2
Where
λ
,
μ
are scalars. Derive the values of
λ
and
μ
such that
{
u
1
,
u
2
,
u
3
}
is an orthogonal basis for the subspace of
V
spanned by
{
u
1
,
u
2
,
v
}
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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