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

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.

d

explanation:

v=uf/(u-f)

= 5.10/(5-10) convex lens has positive focal length

=50/-5

=-10 cm (formed at other side)