Question

**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.

Answer #1

Part 1: We've seen that most of the vector spaces can be
equipped with inner product functions.
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 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.

A2. Let v be a fixed vector in an inner product space V. Let W
be the subset of V consisting of all vectors in V that are
orthogonal to v. In set language, W = { w LaTeX: \in
∈V: <w, v> = 0}. Show that W is a subspace of V. Then,
if V = R3, v = (1, 1, 1), and the inner product is the usual dot
product, find a basis for W.

1. V is a subspace of inner-product space R3,
generated by vector
u =[2 2 1]T and v
=[ 3 2 2]T.
(a) Find its orthogonal complement space V┴ ;
(b) Find the dimension of space W = V+ V┴;
(c) Find the angle θ between u and
v and also the angle β between u
and normalized x with respect to its 2-norm.
(d) Considering v’ =
av, a is a scaler, show the
angle θ’ between u and...

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. 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 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.

Find an inner product on vector space P of polynomials that
makes x and y orthogonal. (x=1+t, y=1-t)

Chapter 6, Section 6.2, Question 16
Let R4 have the Euclidean inner product.
Find two unit vectors that are orthogonal to all three vectors
u = (4, 3, -8, 0), v = (-1, -1,
2, 2), and w = (3, 2, 5, 5). Give the exact
answers in increasing order of the first component.
X1?
(__,__,__,__)
X2?
(__,__,__,__)

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