Let S1 and S2 be subspaces of a vector space V Show that S1

Let S1 and S2 be subspaces of a vector space V . Show that S1 ? S2 is a subspace of V .

Question 5. Let Si and S2 be subspaces of a vector space V. Show that S1n S2 is a subspace of V.

Solution

We can use the subspace theorem to show this. First, since S1 and S2 are both subspaces, they must both contain the zero vector, and therefore ~0 S1 S2, showing that S1 S2 not equal to zero. Suppose that ~v1, ~v2 S1 S2. Then we have that ~v1, ~v2 S1 and ~v1, ~v2 S2. Since S1 and S2 are closed under addition, ~v1 + ~v2 S1 and ~v1 + ~v2 S2 ~v1 + ~v2 S1 S2, so S1 S2 is closed under addition. Also, if r R, then r~v1 S1 and r~v1 S2 r~v1 S1 S2. By the subspace theorem, S1 S2 is a subspace of V