Question

Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V , is an isomorphism.

Answer #1

give an example of a finite dimensional real inner product space
V, an operator T on V and a subspace W of V such that W is a
T-invariant subspace of V. is it possible to find such an example
such that the operator T is self-adjoint?

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.

Let V be an inner product space. Prove that if w⃗ is orthogonal
to each of the vectors in the set
S = {⃗v1, ⃗v2, . . . , ⃗vm}, then w⃗ is also orthogonal to each
of the vectors in the subspace W = SpanS of V .

Suppose that V is a finite dimensional inner product space over
C and dim V = n, let T be a normal linear transformation of V
If S is a linear transformation of V and T has n distinc
eigenvalues such that ST=TS. Prove S is normal.

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 v be an inner product space with an inner product(u,v) prove
that ||u+v||<=||u||+||v||, ||w||^2=(w,w) , for all u,v load to
V. hint : you may use the Cauchy-Schwars inquality: |{u,v}|,=
||u||*||v||.

Let V be a ﬁnite dimensional vector space and T ∈ L(V : V ),
such that, T3 = 0. a) Show that the spectrum of T is σ(T) = {0}. b)
Show that T cannot be diagonalized (unless we are in the trivial
case T = O).

Say (V,<,>) is a finite dimensional real inner product
space. Then the composition of two self adjoint operators is again
self adjoint. Prove or Disprove.

Question 1: Is there a vector space that can
not be an inner product space? Justify your answer.
Part 2: 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.

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.

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