Algebra II MATH 251 Instructor: Dr. E. Goren.

Assignment 7

To be submitted by February 28, 12:00

1. Let W be the subspace of 𝔽^{4} (𝔽 = ℝ or ℂ) defined by the equation x_{1} + x_{2} + x_{3} + x_{4} = 0. Find the orthogonal projection of (1, 0, 0, 0) on W. Remark: I would allow here solutions that use Maple (or other software) if you include a print out of the commands you used.

2. Let V be a vector space with inner-product ⟨·, ·⟩ and associated norm ∥ · ∥.

(1) Prove the parallelogram law

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

(2) We say that u and v are perpendicular (or orthogonal) and write u ⊥ v if ⟨u, v⟩ = 0. Prove

Pythagoras’s Theorem: if u and v are perpendicular then ∥u∥^{2} + ∥v∥^{2} = ∥u + v∥^{2}.

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. Let ( \begin{matrix}a & b \\ c & d \end{matrix} ) be a matrix of complex numbers.

(1) 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.

(2) 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) 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.

1This is also true for ℂ[x]_{n} if we define ⟨f, g⟩ = ∫_{a}^{b} f(x)\overline{g(x)}dx. 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).

6. Given an inner product, define the angle θ between two vectors by

cos(θ) = \frac{⟨u, v⟩}{∥u∥ · ∥v∥}.

This determined θ up to a sign, as one may expect. Show that in the case of u, v ∈ ℝ^{2}, with the standard inner product, this agrees with the geometric definition.

(Bonus question - up to 10 %) \underline{Latin squares}: A Latin square is a square matrix of order n in which each row and each column are permutations of {1, 2, …, n}. Such matrices are important for group theory, experimental designs and linear algebra; there are many open questions. It is a hard theorem that there are more than (n!)^{2n}/n^{n^{2}} Latin squares of order n (this is more than exponential in n).

Let 𝔽 be a finite field with q elements. Fix a line ℓ in the projective plane ℙ^{2}(𝔽) and three distinct points x, y, z on it. Enumerate the lines through x by ℓ_{1}, …, ℓ_{q+1}; the lines through y by m_{1}, …, m_{q+1} and the lines through z by n_{1}, …, n_{q+1}. We may assume that the lines ℓ_{q+1}, m_{q+1} and n_{q+1} are all equal and are the original line on which x, y, z lie.

Define a matrix (a_{ij}) as follows. Consider the point of intersection of ℓ_{i} with m_{j} and the unique line between it and n. This line is some z_{k}. Put a_{ij} = k. Show that this is a well defined matrix that is a Latin square of order q.

Gram-Schmidt on Maple. One can perform the Gram Schmidt process on Maple.

with(LinearAlgebra):

w1 := ⟨2, 1, 0, −1⟩;

w2 := ⟨1, 0, 2, −1⟩;

w3 := ⟨0, −2, 1, 0⟩;

orthobasis :=GramSchmidt([w1,w2,w3]);

orthobasis := [\begin{matrix}2 & 0 & 2/3 \\ 1 & −1/2 & −4/3 \\ 0 & 2 − 1/3 \\ −1 & −1/2 & 0 \end{matrix}]