 # (a) Suppose U and W are subspaces of a vector space V. Prove that U \cap W = \{v : v \in U \text{ widdonod1t 2022-01-04 Answered
(a) Suppose U and W are subspaces of a vector space V. Prove that is a subspace of V.
(b) Give an example of two subspaces U and W and a vector space V such that is not a subspace of V.
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a) Suppose we have two spaces subspaces U and W of V To prove U $\cap$ W is a subspace of V, it is sufficient to prove that any linear combination also belongs to U $\cap$ W.
Let us choose two vectors u and w belong to U $\cap$ W.
i.e. u,w $\in$ U $\cap$ W
i.e. u,w $\in$ U and u,w $\in$ W Since U and W are vector spaces and u,w $\in$ U and u,w $\in$ W,so their linear combination is also belong to that spaces.
i.e. $\alpha u+\beta v\in$ U and $\alpha u+\beta v\in$ W where $\alpha ,\beta$ are any scaler come from underlying scaler field.
i.e. $\alpha u+\beta v\in$ U $\cap$ W It shows that any linear combination of the vectors u and w also belongs to U $\cap$ W.
This prove that U $\cap$ W is the subspace of V. (proved)
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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 $\cup$ W is a subspace of V if and only if either U $\subset$ W or W $\subset$ U.
Now using this we can construct an example.
We know that $\left({\mathbb{R}}^{2}\right)\mathbb{R}$ is a vector space over $\mathbb{R}$.
Let us assume the subspaces of ${\mathbb{R}}^{2}$:
U={(x,y):2x+3y=7} and
W={(x,y):3x+5y=9}
But here neither U $\subset$ W nor W $\subset$ U.
So U $\cup$W is not a subspace of $\left({\mathbb{R}}^{2}\right)\mathbb{R}$ here.