Week 5 Homework (MATH 3130.300)
Summer 2007
HW 19, Due: 7/3
Book Problems 4.5: 4,7,9,11,14,15,19,20,21,23,25,26,29,31
Additional Problems
(1) We have seen that Rnhas dimension n,Pnhas dimension n+ 1, and Pis infinite dimensional.
Lets investigate the dimension of Mm×n. Find a basis for M2×2(prove that what you found is
indeed a basis), and determine the dimension of M2×2. Based on your findings, make an educated
guess about the dimension of Mm×n.
HW 20, Due: 7/5
Book Problems 4.6: 1,2,5,7,10,17,18,22,27,28,29
Additional Problems
(1) Suppose that T:R3→Ris given by T(x) = Axfor Aa 1 ×3 matrix. Describe geometrically
the possibilities for the kernel of T.
HW 21, Due: 7/6
Book Problems 4.7: 1,3,6,7,9,12,13,14,20(a)
Additional Problems
Read the section...again.
HW 22, Due: 7/10
Book Problems 5.1: 1,4,6,11,15,17,18,19,21,23,24,27,32
Additional Problems
(1) Let Abe an n×nmatrix. Suppose that vis an eigenvector of Awith eigenvalue λ(i.e.
Av=λv). Show that for any positive integer k,vis an eigenvector of Ak, and determine the
associated eigenvalue (here Akis the matrix obtained by multiplying Awith itself ktimes).
Challenge Problem
Let Aand Bbe m×nmatrices. Prove that rank(A+B)≤rank A+ rank B .