Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 7

To be submitted by March 16, 12:00 In this assignment V is an inner product space over 𝔽, where 𝔽 = ℝ or ℂ.

1. Prove the Parallelogram Law:

∥u + v∥^{2} + ∥u − v∥^{2} = 2(∥u∥^{2} + ∥v∥^{2}).

2. Let (\begin{matrix}a & b \\ c & d \end{matrix}) be a matrix of complex numbers. Prove that the function

⟨(x_{1}, x_{2}), (y_{1}, y_{2})⟩ = (x_{1}, x_{2}) (\begin{matrix}a & b \\ c & d \end{matrix}) (\overline{y_{1}} ; \overline{y_{2}}) ,

is an inner product on ℂ^{2} if and only if a and d are positive real numbers, c = \overline{b} and ad − b\overline{b} > 0. In the case of the matrix ( \begin{matrix}1 & 1+i \\ 1−i & 5 \end{matrix} ) compute ⟨(1, 2), (3, 4)⟩ and ∥(2, 5i)∥.

3. Let a < b be real numbers. Show that the function

⟨f, g⟩ = ∫_{a}^{b} f(x)g(x)dx

defines an inner product of ℝ[x]_{n}.^{1} Compute the norm of the vector f(x) = 1 + x + x^{2} in the case (a, b) = (0, 1) and in the case (a, b) = (0, 2).

4. Find an orthonormal basis for ℂ^{2} with the inner product defined by the matrix ( \begin{matrix}1 & 1+i \\ 1−i & 5 \end{matrix} ).

5. Perform the Gram-Schmidt process for the basis {1, x, x^{2}} to ℝ[x]_{3} with respect to the inner product

⟨f, g⟩ = ∫_{−1}^{1} f(x)g(x)dx.

6. Let ⟨·, ·⟩ be an inner product on 𝔽^{n}. Define the unit ball in 𝔽^{n} to be the set

𝔹 = {(x_{1}, …, x_{n}) : ∥(x_{1}, …, x_{n})∥ ≤ 1}.

Show that this is a central convex body:

(1) 0 ∈ 𝔹.

(2) if α ∈ [0, 1] and u, v ∈ 𝔹 then αu + (1 − α)v ∈ 𝔹;
(3) if |α| ≤ 1 then v ∈ 𝔹 ⇒ αv ∈ 𝔹.

A difficult question to the valiant among us: Which convex bodies actually arise from inner products?!

⟨f, g⟩ = ∫_{a}^{b} f(x)\overline{g(x)}dx.

1This is also true for ℂ[x]_{n} if we define Note that in this case one can do the integration formally because we are dealing with polynomials. Thus, for example,

⟨x^{2}, 1 + ix⟩ = ∫_{a}^{b}(x^{2} − ix^{3}) dx = (x^{3}/3 − ix^{4}/4)|_{a}^{b} = (b^{3}/3 − ib^{4}/4) − (a^{3}/3 − ia^{4}/4).