Question

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.

.

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 = ___________

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 =

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.

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.

Let B={u1,...un} be an orthonormal basis for inner product space
V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun
where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>

If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and u3 are each a linear combination of them, prove
that {u1, u2, u3} is linearly dependent.
Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . ,
v n } is a basis for a vector space V, then every set
containing
more than n vectors in V is linearly dependent."
Prove without...

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1]
u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]
(a) Explain why these vectors cannot possibly be
independent.
(b) Form a matrix A whose columns are the ui’s and compute the
rref(A).
(c) Solve the homogeneous system Ax = 0 in parametric form and
then in vector form. (Be sure the...

3. a. Consider R^2 with the Euclidean inner product (i.e. dot
product). Let
v = (x1, x2) ? R^2. Show that (x2, ?x1) is orthogonal to v.
b. Find all vectors (x, y, z) ? R^3 that are orthogonal (with
the Euclidean
inner product, i.e. dot product) to both (1, 3, ?2) and (2, 7,
5).
C.Let V be an inner product space. Suppose u is orthogonal to
both v
and w. Prove that for any scalars c and d,...

Let W be an inner product space and v1,...,vn a basis of V. Show
that〈S, T 〉 = 〈Sv1, T v1〉 + . . . + 〈Svn, T vn〉
for S,T ∈ L(V,W) is an inner product on L(V,W).
Let S ∈ L(R^2) be given by S(x1, x2) = (x1 + x2, x2) and let I ∈
L(R^2) be the identity operator. Using the inner product defined in
problem 1 for the standard basis and the dot product, compute 〈S,...

Orthogonalize the basis vectors in the spanning set
p=2x=1 and q=3x+2 with the inner
product of p and q defined to be the evaluation inner product
evaluated at x=-1 and x=2. Use gram Schmidt to orthogonalize.

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