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

Question

Physics

Killian Cline

23 September, 23:35

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

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Answers (1)

Know the Answer?

Ryker Daugherty · Answer 1

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.