Question

Find an inner product such that the vectors ( −1, 2 )T and ( 1, 2...

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.

.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Use the inner product (u, v) = 2u1v1 + u2v2 in R2 and the Gram-Schmidt orthonormalization...
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...
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 =
Use the inner product <u,v>= 2u1v1 + u2v2 in R2 and the Gram-Schmidt orthonormalization process to...
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...
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,...
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...
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...
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 =...
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,...
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...
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,...