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

Assignment 4

To be submitted by February 7, 12:00

1. Let W = {(x, y, z, w) : x + y + z + w = 0, x − y + z − w = 0}, U_{1} = {(x, x, x, x) : x ∈ ℝ} be subspaces of ℝ^{4}. Find a subspace U ⊃ U_{1} such that ℝ^{4} = U ⊕ W. Let T be the projection of ℝ^{4} on U along W. Write the matrix representing T with respect to the standard basis.

2. Let V and W be any two finite dimensional vector spaces over a field 𝔽. Let T : V ⟶ W be a linear map. Prove that there are bases of V and W such that with respect to those bases T is represented by a matrix composed of 0’s and 1’s only. If T is an isomorphism, prove that with respect to suitable bases it is represented by the identity matrix.

3. Let V be a vector space of dimension n and let W be a vector space of dimension m, both over the same field 𝔽. Prove that Hom(V, W) ≅ M_{m×n}(𝔽) as vector spaces over 𝔽. Here M_{m×n}(𝔽) stands for matrices with n columns and m rows with entries in 𝔽.

4. Consider the transformation that rotates the plane ℝ^{2} by angle θ counter-clockwise. Write this transformation as a matrix in the standard basis. Write it also as a matrix with respect to the basis (1, 1), (1, 0).

5. Explain how we may view the complex numbers ℂ as a vector space of dimension 2 over ℝ. Find a basis for ℂ over ℝ. Show that multiplication by i is a linear transformation with respect to the structure of a vector space over ℝ. Write a two-by-two matrix A representing it and prove that A^{2} + I = 0 (here I is the two-by-two identity matrix).

6. Write the following permutation σ as a product of disjoint cycles. For each cycle determine its sign and its order. Determine the sign and order of the permutation σ itself.

σ = (\begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 6 & 4 & 5 & 1 & 2 & 7 & 9 & 8 \end{matrix})

7. For the groups S_{3} and S_{4} write explicitly all the odd transformation and all the even transformations (make a list!).

Bonus question - up to 20 points. What can you say on the maximal order of an element of the symmetric group S_{n}?