Prove that if H and K are subspaces of V then the intersection of H and K is a subspace of V.
I really have no clue on this one.
Thanks in advance!
I'm not going to give you the answer, but...
Think about it this way...
H is a part of V
K is a part of V
The intersection of H and K is both a part of H and of K
What do we know about H and K?
I hope this is enough help, without spelling it out.
I know that a subspace satisfies the following axioms:
a. Contains the zero vector
b. if the vectors u and v are in V u + v must be in V
c. if c is some scalar cu must be in V for all c
So I'm just trying to figure out how to articulate (in a way pleasing to my professor) that since the intersection of H an K is the set of v in V that belong to both H and K, it is a subspace of V.
Proofs are really my weak point (taking the class next semester)...
Basically I am trying to figure out how to write this in "Math language." How do I apply the axioms above to the intersection, or do I even need to?
I know that since H and K are subspaces that the intersection contains the zero vector, but how do I apply the other two rules?
All right, let's right down what we know.
Let h1,h2 be in H, and k1,k2 be in K
Since h1,h2 are in H, then h1+h2 is in H, and so on.
Now, let us look at the vectors in the intersection of H and K.
If the only vector in the intersection is the zero vector, then the other rules are easy to check.
If there are other vectors in the intersection, then you need to check. Remember that if x is in the intersection of H and K, then x is in H and x is in K.
I'm trying to be as arbitrary as possible (for n dimensional vector space).
If I said:
Let span(x1, x2,...,xn) be the set of all vectors on intersection of the subspaces H and K, and c be some scalar then,
c(x1 + x2 +...+ xn) = cx1 + cx2 +...+ cxn = v
will describe some vector v on both H and K, and will thus belong to V.
Would that be valid? I feel like I am saying because the intersection of H and K is in V, the intersection of H and K is in V.
Thanks for the help by the way.
All you really need to say for this part is that H is in V and K is in V therefore the intersection of H and K is in V.I feel like I am saying because the intersection of H and K is in V, the intersection of H and K is in V.
But this only proves that the intersection of H and K is a subset of V. To prove that it is a subspace of V, you also need to show that it satisfies the axioms you posted. ie.
Prove that the 0 vector is in the intersection
Prove that if u and v are in the intersection that u+v is in the intersection
Prove that if c is a scalar then cu is in the intersection
The important thing to remember in proving these is that u is in the intersection if and only if it is in H and K.