Fundamentals of Matrix Algebra, 2011a

1.1 Introducon to Linear Equaons
1. Did we really have to call the red balls “r”? Could we call them “q”?
2. What if we had 60 balls at the start instead of 30?
Let’s look at the first queson. Would the soluon to our problem change if we
called the red balls q? Of course not. At the end, we’d find that q = 10, and we would
know that this meant that we had 10 red balls.
Now let’s look at the second queson. Suppose we had 60 balls, but the other
relaonships stayed the same. How would the situaon and soluon change? Let’s
compare the “orginal” equaons to the “new” equaons.
Original New
r + b + g = 30 r + b + g = 60
r = 2g r = 2g
b = r + g b = r + g
By examining these equaons, we see that nothing has changed except the first
equaon. It isn’t too much of a stretch of the imaginaon to see that we would solve
this new problem exactly the same way that we solved the original one, except that
we’d have twice as many of each type of ball.
A conclusion from answering these two quesons is this: it doesn’t maer what we
call our variables, and while changing constants in the equaons changes the soluon,
they don’t really change the method of how we solve these equaons.
In fact, it is a great discovery to realize that all we care about are the constants and
the coefficients of the equaons. By systemacally handling these, we can solve any
set of linear equaons in a very nice way. Before we go on, we must first define what
a linear equaon is.
.
Ḋefinion 1
Linear Equaon
A linear equaon is an equaon that can be wrien in the
form
a 1 x 1 + a 2 x 2 + · · · + a n x n = c
.
where the x i are variables (the unknowns), the a i are
coefficients, and c is a constant.
A system of linear equaons is a set of linear equaons that
involve the same variables.
A soluon to a system of linear equaons is a set of values
for the variables x i such that each equaon in the system is
sasfied.
So in Example 1, when we answered “how many marbles of each color are there?,”
we were also answering “find a soluon to a certain system of linear equaons.”
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