Question

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 .” If the statement is true, prove it. If the statement is false, give an example demonstrating this.

Answer #1

Let U and W be subspaces of a nite dimensional vector space V
such that U ∩ W = {~0}. Dene their sum U + W := {u + w | u ∈ U, w ∈
W}.
(1) Prove that U + W is a subspace of V .
(2) Let U = {u1, . . . , ur} and W = {w1, . . . , ws} be bases
of U and W respectively. Prove that U ∪ W...

Let S, U, and W be subspaces of a vector space V, where U ⊆ W.
Show that U + (W ∩ S) = W ∩ (U + S)

Let U and W be subspaces of a finite dimensional vector space V
such that V=U⊕W. For any x∈V write x=u+w where u∈U and w∈W. Let
R:U→U and S:W→W be linear transformations and define T:V→V by
Tx=Ru+Sw
.
Show that detT=detRdetS
.

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.

Let V be a vector space and let U1, U2 be two subspaces of V .
Show that U1 ∩ U2 is a subspace of V . By giving an example, show
that U1 ∪ U2 is in general not a subspace of V .

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.

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 be an n-dimensional vector space. Let W and W2 be unequal
subspaces of V, each of dimension n - 1. Prove that V =W1 + W2 and
dim(Win W2) = n - 2.

Suppose U and W are subspaces of V. Prove that U+W is a subspace
of V.

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