(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 UW={v:vUand vW} is a subspace of V.
(b) Give an example of two subspaces U and W and a vector space V such that UW={v:vUor vW} is not a subspace of V.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

vrangett
Answered 2022-01-05 Author has 36 answers
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)
Not exactly what you’re looking for?
Ask My Question
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 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.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more