# Math 296 problems 6

Igor Kriz

Regular problems:

1.

Let

(a) Find a basis of Null(A).

(b) Find a basis of Col(A).

2.

Review problem: Find the shortest distance from a given point (0,b) on the y-axis to the parabola . [Express the distance as a function, and find its minimum using derivatives.]

3.

Let V be a vector space, let S and T be two subsets of V (not necessarily subspaces).

(a) Prove that .

(b) Find an example where

4. Consider the set

Does S span ? Is S linearly independent in ?

Challenge problems:

5.

Linear recursions continued: Suppose a sequence is defined as follows: ,..., are given, and

(a) Suppose that the polynomial

has a root of multiplicity (i.e. divides p(x)). Show that then the sequences

for i;SPMlt;k-1 and all their linear combinations satisfy the relation (1). [Hint: use derivatives.]

(b) Using (a), solve the following problem: Suppose numbers are given as follows: , , for . Find a formula for .

(c) Consider the sequence 1,4,2,1,4,2,1,4,2,.... Thus, , and for all natural numbers n. Find a formula for which uses only arithmetic operations (addition, multiplication, subtraction, division, taking powers and roots). [Write the sequence in terms of a linear recursion. This does not use (a) or (b), but it uses complex numbers.]

6.

Review problem: If

where are real constants, prove that the equation

has at least one real solution .

Igor Kriz
Tue Feb 10 12:12:07 EST 1998