Page 27 - 35Linear Algebra

1.4 So, What is a Matrix? 27

2
2
= (2ax + b) + (2ax + 2bx + 2c) = 2ax + (2a + 2b)x + (b + 2c)

     
2a 2 0 0 a
b
=  2a + 2b  =  2 2 0    .
b + 2c 0 1 2 c
B B

That is, our notational convention for quadratic functions has induced a notation for
the differential operator d + 2 as a matrix. We can use this notation to change the
dx
way that the following two equations say exactly the same thing.

     
2 0 0 a 0

d
+ 2 f = x + 1 ⇔  2 2 0    =   .
b
1
dx
0 1 2 c 1
B B
Our notational convention has served as an organizing principle to yield the system of
equations
2a = 0
2a + 2b = 1
b + 2c = 1

0
 
1
with solution   , where the subscript B is used to remind us that this stack of
2
1
4 B
1
1
numbers encodes the vector x+ , which is indeed the solution to our equation since,
2 4
1
1

substituting for f yields the true statement d + 2 ( x + ) = x + 1.
dx 2 4
It would be nice to have a systematic way to rewrite any linear equation
as an equivalent matrix equation. It will be a little while before we can learn
to organize information in a way generalizable to all linear equations, but
keep this example in mind throughout the course.

The general idea is presented in the picture below; sometimes a linear
equation is too hard to solve as is, but by organizing information and refor-
mulating the equation as a matrix equation the process of finding solutions
becomes tractable.

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