Question

Determine whether the given set ?S is a subspace of the vector
space ?V.

**A.** ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.

**B.** ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.

**C.** ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).

**D.** ?=?3(?)V=C3(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?‴+7?=?2.y‴+7y=x2.

**E.** ?=??(ℝ)V=Mn(R), and ?S is the subset of all
symmetric matrices

**F.** ?=ℝ?V=Rn, and ?S is the set of solutions to the
homogeneous linear system ??=0Ax=0 where ?A is a fixed ?×?m×n
matrix.

**G.** ?=ℝ2V=R2, and ?S consists of all vectors
(?1,?2)(x1,x2) satisfying ?21−?22=0.x12−x22=0.

Answer #1

Determine if the given set V is a subspace of the vector space
W, where
a) V={polynomials of degree at most n with p(0)=0} and W=
{polynomials of degree at most n}
b) V={all diagonal n x n matrices with real entries} and W=all n
x n matrices with real entries
*Can you please show each step and little bit of an explanation
on how you got the answer, struggling to learn this concept?*

A vector space V and a subset S are given.
Determine if S is a subspace of V by determining
which conditions of the definition of a subspace are satisfied.
(Select all that apply.)
V = C[−4, 4] and S = P.
S contains the zero vector.
S is closed under vector addition.
S is closed under scalar multiplication.
None of these

Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.

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}...

Definition. Let S ⊂ V be a subset of a vector space. The span of
S, span(S), is the set of all finite
linear combinations of vectors in S. In set notation,
span(S) = {v ∈ V : there exist v1, . . . , vk ∈ S and a1, . . . ,
ak ∈ F such that v = a1v1 + . . . + akvk} .
Note that this generalizes the notion of the span of a...

For a nonempty subset S of a vector space V , define span(S) as
the set of all linear combinations of vectors in S.
(a) Prove that span(S) is a subspace of V .
(b) Prove that span(S) is the intersection of all subspaces that
contain S, and con- clude that span(S) is the smallest subspace
containing S. Hint: let W be the intersection of all subspaces
containing S and show W = span(S).
(c) What is the smallest subspace...

Let S be a set in a vector space V and v any vector. Prove that
span(S) = span(S ∪ {v}) if and only if v ∈ span(S).

Prove that the singleton set {0} is a vector subspace of the space
P4(R) of all polynomials of degree at most 3 with real
coefficients.

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 S={u,v,w}S={u,v,w} be a linearly independent set in a vector
space V. Prove that the set S′={3u−w,v+w,−2w}S′={3u−w,v+w,−2w} is
also a linearly independent set in V.

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