Question

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.

Answer #1

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...

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?
(__,__,__,__)

We have learned that we can consider spaces of matrices,
polynomials or functions as vector spaces. For the following
examples, use the definition of subspace to determine whether the
set in question is a subspace or not (for the given vector space),
and why.
1. The set M1 of 2×2 matrices with real entries such that all
entries of their diagonal are equal. That is, all 2 × 2 matrices of
the form: A = a b c a
2....

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

Vector B is formed by the product of two gradients B = (∇u) ×
(∇v) , where u and v are scalar functions. Show that:
(a) B is solenoidal;
(b) A = 1/2(u∇v − v∇u) is a vector potential for B,
that is B =∇×A

Suppose V and W are two vector spaces. We can make the set V × W
= {(α, β)|α ∈ V,β ∈ W} into a vector space as follows:
(α1,β1)+(α2,β2)=(α1 + α2,β1 + β2)
c(α1,β1)=(cα1, cβ1)
You can assume the axioms of a vector space hold for V × W
(A) If V and W are finite dimensional, what is the dimension of
V × W? Prove your answer.
Now suppose W1 and W2 are two subspaces of V ....

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