(a) Let U and W be subspaces of a vector space V. Prove that &nbsp;U∩W={v:v∈Uand&nbsp;v∈W}&nbsp;is a subspace of V.&nbsp;(b) Give an example of two subspaces U and W and a vector space V such that&nbsp;U∪W={v:v∈Uor&nbsp;v∈W}&nbsp;is not a subspace of V.

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(a) Let U and W be subspaces of a vector space V. Prove that &amp;nbsp;U∩W={v:v∈Uand&amp;nbsp;v∈W}&amp;nbsp;is a subspace of V.&amp;nbsp;(b) Give an example of two subspaces U and W and a vector space V such that&amp;nbsp;U∪W={v:v∈Uor&amp;nbsp;v∈W}&amp;nbsp;is not a subspace of V.

vrangett · Accepted Answer

a) Suppose we have two  spaces subspaces  U and W of V To prove U ∩ W is a  subspace of V, it is sufficient to prove that any linear combination also belongs to U ∩ W.   Let us choose two vectors u and w belong to U ∩ W.  i.e. u,w ∈ U ∩ W   i.e.  u,w ∈ U and u,w ∈ W Since U and W are vector spaces and u,w ∈ U and u,w ∈ W,so their linear combination is also belong to that spaces.  i.e. αu+βv∈ U and αu+βv∈ W where α,β are any scaler come from underlying scaler field.  i.e. αu+βv∈ U ∩ W It shows that any linear combination of the vectors u and w also belongs to U ∩ W.  This prove that U ∩ W is the subspace of V. (proved)

Kayla Kline · Accepted Answer

b) To give an example of two  vector spaces whose union is not a vector space we have know the following which give the proper scenario when the union of two vector spaces is again a subspace of the vector spaces.   Suppose U and W are to subspaces. Then U  ∪ W is a  subspace  of V  if and only if either U ⊂ W or W ⊂ U.   Now using this we can construct an example.   We know that (R2)R is a vector space over R.   Let us assume the subspaces of R2:   U={(x,y):2x+3y=7} and    W={(x,y):3x+5y=9}  But here neither U ⊂ W nor W ⊂ U.   So  U  ∪W is not a subspace of (R2)R here.

(a) Suppose U and W are subspaces of a vector space V. Prove that U \cap W = \{v : v \in U \text{

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