Page 21 - 35Linear Algebra

1.4 So, What is a Matrix? 21

Each bag contains 2 apples and 4 bananas and each box contains 6 apples and 8
bananas. There are 20 apples and 28 bananas in the room. Find x and y.

The values are the numbers x and y that simultaneously make both of the following
equations true:

2 x + 6 y = 20
4 x + 8 y = 28 .
8
Here we have an example of a System of Linear Equations. It’s a collection
of equations in which variables are multiplied by constants and summed, and
no variables are multiplied together: There are no powers of variables (like x 2
5
−3
or y ), non-integer or negative powers of variables (like y 1/7 or x ), and no
places where variables are multiplied together (like xy).
Reading homework: problem 1

Information about the fruity contents of the room can be stored two ways:
(i) In terms of the number of apples and bananas.

(ii) In terms of the number of bags and boxes.

Intuitively, knowing the information in one form allows you to figure out the
information in the other form. Going from (ii) to (i) is easy: If you knew
there were 3 bags and 2 boxes it would be easy to calculate the number
of apples and bananas, and doing so would have the feel of multiplication
(containers times fruit per container). In the example above we are required
to go the other direction, from (i) to (ii). This feels like the opposite of
multiplication, i.e., division. Matrix notation will make clear what we are
“multiplying” and “dividing” by.
The goal of Chapter 2 is to efficiently solve systems of linear equations.
Partly, this is just a matter of finding a better notation, but one that hints
at a deeper underlying mathematical structure. For that, we need rules for
adding and scalar multiplying 2-vectors;
0
0
x cx x x x + x
c := and + := .
y cy y y 0 y + y 0

x
8 Perhaps you can see that both lines are of the form Lu = v with u = an unknown,
y
v = 20 in the first line, v = 28 in the second line, and L different functions in each line?
We give the typical less sophisticated description in the text above.

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