Question

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 containing the empty set ;?

Answer #1

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...

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}...

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.

proof of the following statement
If S= {v1, v2, ...., vr} is a nonempty set of vectors in a
vector space V then : W is the smallest subspace of V that contains
all of the vectors in S in the sense that any other subspace of V
that contains those vectors must contain W.

For any subset S ⊂ V show that span(S) is the smallest subspace
of V containing S. (Hint: This is asking you to prove several
things. Look over the proof that U1+. . .+Um is the smallest
subspace containing U1, . . . , Um.)

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...

9. Let S and T be two subspaces of some vector space V.
(b) Define S + T to be the
subset of V whose elements have the form (an
element of S) + (an element of
T). Prove that S +
T is a subspace of V.
(c) Suppose {v1, . . . ,
vi} is a basis for the
intersection S ∩ T. Extend this with
{s1, . . . ,
sj} to a basis for
S, and...

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).

Let U1, U2 be subspaces of a vector space V.
Prove that the union of U1 and U2 is a subspace if and only if
either U1 is a subset of U2 or U2 is a subset of U1.

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.

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