1) Let V be an inner product space and T:V ® V a linear transformation

1) Let V be an inner product space and T:V ® V a linear transformation. Suppose that

                  (T(v), T(v)) £ (v, v)   for all v Î V.

Prove that the linear transformation T - Ö3 id_v from V to V is injective.

2) Let V be an inner product space and u, v Î V be vectors such that

                           (u, u) = (v, v) = (u, v) = 1.

Prove that u = v.

3) Suppose that x, y, z, w, t are positive real numbers. Prove that

           (x + y + z + w + t) (1/x + 1/y + 1/z + 1/w + 1/t) ³ 25

4) Let V be an inner product space and u, v Î V vectors such that

                |u| = 2,      |u + v| = 3,   |u – v|= 4     

             Find |v|.

5) Consider a triangle with side lengths a, b, c. Let m denote the length of the median of the side of length c (this is the liner segment joining the vertex opposite to the side of length c to the midpoint of the side). Prove that

            a² + b² = ½ c² + 2m²

6) Let V be an inner product space, and let {u₁, …, u_m} be an orthonormal subset of V. Let v Î V be an arbitrary vector. Show that v Î Span{u₁, ..,u_m} if and only if

         |v|² = |(v, u₁)|² + … +|(v, u_m)|²