(a) Find a basis of Null(A).
(b) Find a basis of Col(A).
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.]
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 ?
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.]
Review problem: If
where are real constants, prove that the equation
has at least one real solution .