MATH 211 Written Homework # 3 (due Friday, 1 November 2019)
1. Given an m × n matrix A, define a new matrix A in which each entry of A becomes a 2 × 2 submatrix in A with that same entry. Prove that the nullity of A equals n plus the nullity of A. (As an example, if
then we get
1 2 2 A=345
1 1 2 2 2 2 A=1 1 2 2 2 2.
3 3 4 4 5 5 334455
The nullity of A is 1 and the nullity of A is 4.)
2. Let A = Jnn, the n × n all-ones matrix.
(a) Give the elementary matrices used to reduce A to RREF. (b) Find an LU-decomposition of A.
3. Showthatthesetofpolynomials{1,x−1,(x−1)2,…,(x−1)k}islinearlyindependent.
4. ConsiderthesetS={A∈M(3,3):A⊤ =−A}.
(a) Show that S is a subspace of M(3,3). (b) Find a basis for S.
5. Consider R2 with the usual vector addition and the following strange scalar multipli- cations (a), (b), (c) shown below. Show that none of these form a vector space over R by identifying a problematic axiom from the list V1-V10. Give explicit values to show the axiom fails.
(a) c∗(x1,x2)=(0,cx2)
(b) c⋆(x1,x2)=(cx1,c2x2)
(x1, x2) if c = 1,
(c) c (x1, x2) = (cx2, cx1) otherwise.