Question

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 .

Answer #1

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.

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)

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}...

T/ F : Let V be an inner product space with orthogonal basis B =
{v1, . . . , vn}. Let [v]B = (1, 2, 2, 0, . . . , 0). Then ||v|| =
3.
The ans is F , but I don't understand why. Please explain.

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.

Prove that
Let S={v1,v2,v3} be a linearly indepedent set of vectors om a
vector space V. Then so are
{v1},{v2},{v3},{v1,v2},{v1,v3}，{v2,v3}

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
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.

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.

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,...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 34 minutes ago

asked 58 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago