Let U and V be subspaces of a vector space W. Prove that their intersection is also a subspace of W
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Let X be the intersection of U and V.
1) X closed under addition:
Assume x in X and y in X. From this, we know x in U and y in U.
But since we know that U is a subspace, x+y in U holds. Similarly,
one can show that x+y in V and therefore x+y in X. So X is indeed
closed under addition.
2) X closed under scalar multiplication:
Assume x in X and r in R. Again, we know x in U. Since U is a
subspace, it is closed under scalar multiplication. Therefore, r*x in
U holds. Also r*x in V holds with a similar argument. From this r*x
in X follows. So X is closed under scalar multiplication.
Therefore by subspace theorem, we're done. You can use the definitely of a vectorspace as well, but that's more tedious.