Question

5. Let U_{1}, U_{2}, U_{3} be subspaces
of a vector space V.

Prove that U_{1}, U_{2}, U_{3} are
direct-summable if and only if

(i) the intersection of U_{1} and U_{2} is 0.\,
and

(ii) the intersection of U_{1}+U_{2} and
U_{3} is 0.

A detailed explanation would be greatly appreciated :)

Answer #1

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.

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 .

If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and u3 are each a linear combination of them, prove
that {u1, u2, u3} is linearly dependent.
Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . ,
v n } is a basis for a vector space V, then every set
containing
more than n vectors in V is linearly dependent."
Prove without...

Determine if the vector v is a linear combination of the vectors
u1, u2, u3. If yes, indicate at least one possible value for the
weights. If not, explain why.
v =
2
4
2
, u1 =
1
1
0
, u2 =
0
1
-1
, u3 =
1
2
-1

Consider the vector u1=(2,0,2), u2=(4,1,-1), u3=( 0,1,-5),
u4=(3,0,2)
a) Find the dimension and a basis for U= span{ u1,u2,u3,u4}
b) Does the vector u=(2,-1,4) belong to U. Justify!
c) Is it true that U = span{ u1,u2,u3} justify the answer!

Let B={u1,...un} be an orthonormal basis for inner product space
V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun
where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>

Question 6
Suppose u and v are vectors in 3–space where u = (u1, u2, u3)
and v = (v1, v2, v3).
Evaluate u × v × u and v × u × u.

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...

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...

Consider the vector: u1=(1,1,1), u2= (2,-1,1), u3=(3,0,2),
u4=(6,0,4)
a)Plot the dimension and a basis for W= span {u1,u2,u3,u4}
b)Does the vector v= (3,3,1) belong to W? justify the
answer.
c) Is it true that W= span{ u3,u4} Justify answer.

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