Question

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉.

(b) Suppose {a1, · · · ak} are orthonormal vectors in R m.

Show that {a1, · · · ak} is a linearly independent set

Answer #1

Suppose that u and v are two non-orthogonal vectors in an inner
product space V,< , >.
Question 2: Can we modify the inner product < , > to a new
inner product so that the two vectors become orthogonal? Justify
your answer.

Enlarge the following set to linearly independent vectors to
orthonormal bases of R^3 and R^4
{(1,1,1)^t, (1,1,2)^t}
could you show me the process, please

Let (u,v,w,t) be a linearly independent list of vectors in R4.
Determine if (u, v-u, w+5v, t) is a linearly independent list.
Explain your reasoning and Show work.

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 T be a linear transformation that is one-to-one, and u, v be
two vectors that are linearly independent. Is it true that the
image vectors T(u), T(v) are linearly independent? Explain why or
why not.

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.

5. a) Suppose that the area of the parallelogram spanned by the
vectors ~u and ~v is 10. What is the area of the parallogram
spanned by the vectors 2~u + 3~v and −3~u + 4~v ?
(b) Given (~u × ~v) · ~w = 10. What is ((~u + ~v) × (~v + ~w)) ·
( ~w + ~u)? [4]
6. Find an equation of the plane that is perpendicular to the
plane x + 2y + 4 =...

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 = ___________

Suppose 〈 , 〉 is an inner product on a vector space V . Show
that no vectors u and v exist such that
∥u∥ = 1, ∥v∥ = 2, and 〈u, v〉 = −3.

Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.

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