Question

True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 43. (a) If W is a subspace of a vector space V, then it has closure under scalar multiplication as defined in V. (b) If V and W are both subspaces of a vector space U, then the intersection of V and W is also a subspace. (c) If U, V, and W are vector spaces such that W is a subspace of V and U is a subspace of V, then W = U

Answer 1

Answer:

a)True.

b)True.

c)False

Explanation:

a)

Yes it is true because W is also  a vector  and that is why it satisfy all the principle of a vector spacer.

b)

Yes it is true.

Lets take v have two subspace U₁ and U₂ then U₁ ∩ U₂ will be a subset of V.

Lets a , b ∈ U₁ ∩ U₂ and ∝ ∈ F

1)

a , b ∈ U₁  ⇒ a +b ∈ U₁  then ∝a ∈ U₁

2)

a , b ∈ U₂ ⇒ a +b ∈ U₂  then ∝a ∈ U₁

So we can say that

a +b ∈ U₁ ∩ U₂  and ∈ U₁ ∩ U₂  

So  U₁ ∩ U₂  is a subspace.

c)

It is false.

The two subspace W and U of a vector space V can only same when W=U.When dimensions of W and U will be same only when they will be equal.