Can I have help with the following question related to the concept vector 'subspaces'?
Question : Let X and Y be vector subspaces of a vector space V over a field F, (V,+,•,F).
Prove that the intersection of X and Y is also a vector subspace of V.
(1)So when we take any two vectors u,v€(X^Y) (€denote belongs to a set ans ^ denote intersection),
Then we can show that the vector u+v€ (X^Y)
2) If we consider any scalar A€F, any vector u€(X^Y),
we can show that the vector Au€(X^Y)
My question is, do we need to show that the identity element of V(say 0), also belong to X^Y, to show that X^Y is a vector subspace of V?
Or is proving the conditions (1) & (2) above would be sufficient to prove that X^Y is a vector subspace of V?
Thank you!
To prove that the intersection of X and Y is a vector subspace of V, you need to show that it satisfies the three conditions of a subspace. These conditions are:
1) Non-empty: The subspace must contain at least one vector.
2) Closed under vector addition: If u and v both belong to the subspace, then their sum u + v must also belong to the subspace.
3) Closed under scalar multiplication: If u belongs to the subspace and A is any scalar, then the scalar multiple Au must also belong to the subspace.
Now, let's address your specific questions:
1) To prove that the intersection of X and Y is closed under vector addition, you need to show that for any two vectors u and v that belong to the intersection (u, v ∈ X ∩ Y), their sum (u + v) also belongs to the intersection. This can be done by showing that u and v both belong to X and Y individually, using the definition of intersection.
2) To prove that the intersection of X and Y is closed under scalar multiplication, you need to show that for any scalar A and vector u that belongs to the intersection (A ∈ F, u ∈ X ∩ Y), their scalar multiple (Au) also belongs to the intersection. Again, you can use the definition of intersection to show that u belongs to both X and Y.
3) Regarding the identity element (usually denoted as 0) belonging to the intersection (X ∩ Y), this is not necessary to prove that the intersection is a subspace. The identity element 0 is already a part of the larger vector space V, and since both X and Y are subspaces of V, they already contain the identity element. Thus, the intersection X ∩ Y also automatically contains the identity element 0.
So, to summarize, in order to prove that the intersection of X and Y is a vector subspace of V, you need to show that it satisfies conditions (1) and (2) you mentioned above: closure under vector addition and scalar multiplication. Additionally, it is not necessary to separately show that the identity element belongs to the intersection, as it is already a part of both X and Y as subspaces of V.