Prove that the determinant of an upper triangular matrix is the product of the diagonal entries. Let V be a nite dimensional vector space and let S...

1. Prove that the determinant of an upper triangular matrix is the product of the diagonal entries.

2. Let V be a finite dimensional vector space and let S be a spanning set for V . Prove that a subset B of S can be chosen which is a basis for V . (Hint: Let n = |S| − dim V ≥ 0 and prove by induction on n)

3. Prove that a linear transformation L is injective if and only if ker(L) is the trivial vector space.