Question

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.

Answer #1

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 .

Let W be a subspace of a f.d. inner product space V and let PW
be the orthogonal projection of V onto W. Show that the
characteristic polynomial of PW is
(t-1)^dimW t^(dimv-dimw)

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,...

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...

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

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,...

Let U and V be subspaces of the vector space W . Recall that U ∩
V is the set of all vectors ⃗v in W that are in both of U or V ,
and that U ∪ V is the set of all vectors ⃗v in W that are in at
least one of U or V
i: Prove: U ∩V is a subspace of W.
ii: Consider the statement: “U ∪ V is a subspace of W...

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.

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...

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.

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