Question

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

Answer #1

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}...

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...

use the subspace theorem ( i) is it a non-empty space? ii) is it
closed under vector addition? iii)is it closed under scalar
multiplication?) to decide whether the following is a real vector
space with its usual operations:
the set of all real polonomials of degree exactly n.

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...

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?*

Prove that the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

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...

3. Closure Properties
(a) Using that vector spaces are closed under scalar
multiplication, explain why if any nonzero vector from R2 or R3 is
in a vector space V, then an entire line’s worth of vectors are in
V.
(b) Why isn’t closure under vector addition enough to make the same
statement?
4. Subspaces and Spans: The span of a set of vectors from Rn is
always a subspace of Rn. This is relevant to the problems below
because the...

a)Suppose U is a nonempty subset of the vector space V over
field F. Prove that U is a subspace if and only if cv + w ∈ U for
any c ∈ F and any v, w ∈ U
b)Give an example to show that the union of two subspaces of V
is not necessarily a subspace.

The trace of a square n×nn×n matrix A=(aij)A=(aij) is the sum
a11+a22+⋯+anna11+a22+⋯+ann of the entries on its main
diagonal.
Let VV be the vector space of all 2×22×2 matrices with real
entries. Let HH be the set of all 2×22×2 matrices with real entries
that have trace 11. Is HH a subspace of the vector space
VV?
Does HH contain the zero vector of VV?
choose H contains the zero vector of V H does not contain the zero
vector...

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