**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 .

Tue Feb 10 12:12:07 EST 1998