Question

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} = {all linear combinations of the vi}

We can generalize this definition for any subset S ⊂ V: Span(S) := {all linear combinations of vectors in S}

Prove that in fact Span(S) is always a subspace of V when S is non-empty (i.e. it is true even if S is not necessarily finite).

c. Suppose {v1,. . . , vn} is a finite subset of a vector space V which is linearly independent.

i.Prove that for any i, {v1, . . . , vn} \ {vi} is also a linearly independent set (as long as n>1).

ii.Prove that if {v1, . . . , vn} is not a basis for V then you can find a vector w ∈ V such that {v1, . . . , vn, w} is a linearly independent 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...

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 V and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

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.

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

Let V be a vector space and let v1,v2,...,vn be elements of V .
Let W = span(v1,...,vn). Assume v ∈ V and ˆ v ∈ V but v / ∈ W and ˆ
v / ∈ W. Deﬁne W1 = span(v1,...,vn,v) and W2 = span(v1,...,vn, ˆ
v). Prove that either W1 = W2 or W1 ∩W2 = W.

let T:V to W be a linear transdormation of vector
space V and W and let B=(v1,v2,...,vn) be a basis for V. Show that
if (Tv1,Tv2,...,Tvn) is linearly independent, thenT is
injecfive.

Let u, vand w be linearly dependent vectors in a vector space V.
Prove that for any vector z in V whatsoever, the vectors u, v, w
and z are linearly dependent.

5. Prove or disprove the following statements.
(a) Let L : V → W be a linear mapping. If {L(~v1), . . . , L(
~vn)} is a basis for W, then {~v1, . . . , ~vn} is a basis for
V.
(b) If V and W are both n-dimensional vector spaces and L : V →
W is a linear mapping, then nullity(L) = 0.
(c) If V is an n-dimensional vector space and L : V →...

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...

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