Page 132 - 35Linear Algebra

132 Matrices

8. This exercise is meant to show you a generalization of the procedure
you learned long ago for finding the function mx+b given two points on
its graph. It will also show you a way to think of matrices as members
of a much bigger class of arrays of numbers.

Find the

(a) constant function f : R → R whose graph contains (2, 3).
(b) linear function h : R → R whose graph contains (5, 4).

(c) first order polynomial function g : R → R whose graph contains
(1, 2) and (3, 3).

(d) second order polynomial function p : R → R whose graph contains
(1, 0), (3, 0) and (5, 0).

(e) second order polynomial function q : R → R whose graph contains
(1, 1), (3, 2) and (5, 7).

(f) second order homogeneous polynomial function r : R → R whose
graph contains (3, 2).

(g) number of points required to specify a third order polynomial
R → R.
(h) number of points required to specify a third order homogeneous
polynomial R → R.

(i) number of points required to specify a n-th order polynomial R →
R.
(j) number of points required to specify a n-th order homogeneous
polynomial R → R.

2
(k) first order polynomial function F : R → R whose graph contains

0 0 1 1
, 1 , , 2 , , 3 , and , 4 .
0 1 0 1
2
(l) homogeneous first order polynomial function H : R → R whose

0 1 1
graph contains , 2 , , 3 , and , 4 .
1 0 1
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