Page 133 - 35Linear Algebra

7.3 Properties of Matrices 133

2
(m) second order polynomial function J : R → R whose graph con-

0 0 0
tains , 0 , , 2 , , 5 ,
0 1 2

1 2 1
, 3 , , 6 , and , 4 .
0 0 1
2
2
(n) first order polynomial function K : R → R whose graph con-

0 1 0 2
tains , , , ,
0 1 1 2

1 3 1 4
, , and , .
0 3 1 4
(o) How many points in the graph of a q-th order polynomial function
n
n
R → R would completely determine the function?
(p) In particular, how many points of the graph of linear function
n
n
R → R would completely determine the function? How does a
matrix (in the standard basis) encode this information?
(q) Propose a way to store the information required in 8g above in an
array of numbers.

(r) Propose a way to store the information required in 8o above in an
array of numbers.

7.3 Properties of Matrices

The objects of study in linear algebra are linear operators. We have seen that
linear operators can be represented as matrices through choices of ordered
bases, and that matrices provide a means of efficient computation.
We now begin an in depth study of matrices.

i
Definition An r × k matrix M = (m ) for i = 1, . . . , r; j = 1, . . . , k is a
j
rectangular array of real (or complex) numbers:
 1 1 1 
m m · · · m
1 2 k
 m 2 m 2 · · · m 

2
k
1
2
 . . . . . . 
M =  . . .  .
m r m r · · · m r
1 2 k
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