(b) Give an example of two subspaces U and W and a vector space V such that

widdonod1t
2022-01-04
Answered

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

(b) Give an example of two subspaces U and W and a vector space V such that$U\cup W=\{v:v\in U\text{or}\text{}v\in W\}$ is not a subspace of V.

(b) Give an example of two subspaces U and W and a vector space V such that

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vrangett

Answered 2022-01-05
Author has **36** answers

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)

Let us choose two vectors u and w belong to U

i.e. u,w

i.e. u,w

i.e.

i.e.

This prove that U

Kayla Kline

Answered 2022-01-06
Author has **37** answers

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.

Suppose U and W are to subspaces. Then U

Now using this we can construct an example.

We know that

Let us assume the subspaces of

U={(x,y):2x+3y=7} and

W={(x,y):3x+5y=9}

But here neither U

So U

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