Question

Use the inner product <u,v>= 2u1v1 + u2v2 in R2 and the Gram-Schmidt orthonormalization process to transform {(2, ?1), (2, 6)} into an orthonormal basis. (Use the vectors in the order in which they are given.)

u1 =

u2 =

Answer #1

Use the inner product (u, v) =
2u1v1 +
u2v2 in
R2 and the Gram-Schmidt orthonormalization
process to transform {(?2, 1), (2, 5)} into an orthonormal basis.
(Use the vectors in the order in which they are given.)
u1 = ___________
u2 = ___________

Let
R4
have the inner product
<u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4
(a)
Let w = (0, 6,
4, 1). Find ||w||.
(b)
Let W be the
subspace spanned by the vectors
u1 = (0, 0, 2,
1), and u2 = (3, 0, −2,
1).
Use the Gram-Schmidt process to transform the basis
{u1,
u2} into an
orthonormal basis {v1,
v2}. Enter the
components of the vector v2 into the
answer box below, separated with commas.

Apply the Gram-Schmidt orthonormalization process to transform
the given basis for into an orthonormal basis„ Use the vectors in
the order in which they are given.
B={1,1,1>,<-1,1,0>,<1,2,1>

Apply the Gram-Schmidt orthonormalization process to transform
the given basis for
ℝ4 into an orthonormal basis. Use the vectors in the
order in which they are given.
B ={(3,4,0,0),(−1,1,0,0),(2,1,0,−1),{0,1,1,0}}

Use Gram-Schmidt process to transform the basis {u1, u2, u3},
where u1=(1,1,1), u2=(1,2,0), u3=(1,0,-1),:
a) for the Euclidean IPS. (IPS means inner product space)

Use the Gram-Schmidt process to find an orthonormal basis for
the subspace of R4 spanned by the vectors
u1 = (1, 0, 0, 0), u2 = (1, 1, 0, 0),
u3 = (0, 1, 1, 1).
Show all your work.

Let W be the subspace of R4
spanned by the vectors
u1 = (−1, 0, 1, 0),
u2 = (0, 1, 1, 0), and
u3 = (0, 0, 1, 1).
Use the Gram-Schmidt process to transform the basis
{u1, u2,
u3} into an orthonormal basis.

(a) Use the Gram-Schmidt process on the basis {(1, 2, 2),(1, 2,
3),(4, 3, 2)} of R ^3 find an orthonormal basis.
(b) Write the vector v = (2, 1, −5) as a linear combination of
the orthonormal basis vectors found in part (a).

Suppose u = (u1,u2) and v = (v1, v2) are two vectors in R2.
Explain why the operations (u * v) = u1v2 cannot be an inner
product.

Find an inner product such that the vectors ( −1, 2 )T and ( 1,
2 )T form an orthonormal basis of R2
2
4.1.11. Prove that every orthonormal basis of R2 under the
standard dot product has the form u1 =
cos θ
sin θ
and u2 = ±
− sin θ
cos θ
for some 0 ≤ θ < 2π and some choice of ± sign.
.

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