Linear Algebra

Section I. Solving Linear Systems 13
2.4 Example After this reduction
2x − 2y =0
z +3w =2
3x − 3y =0
x − y +2z +6w =4
−(3/2)ρ1+ρ3
−→
−(1/2)ρ1+ρ4
−2ρ2+ρ4
−→
2x − 2y =0
z +3w =2
0=0
2z +6w =4
2x − 2y =0
z +3w =2
0=0
0=0
x and z lead, y and w are free. The solution set is {(y, y, 2 − 3w, w) � � y, w ∈ R}.
For instance, (1, 1, 2, 0) satisfies the system — take y =1andw =0. The
four-tuple (1, 0, 5, 4) is not a solution since its first coordinate does not equal its
second.
We refer to a variable used to describe a family of solutions as a parameter
and we say that the set above is paramatrized with y and w. (The terms
‘parameter’ and ‘free variable’ do not mean the same thing. Above, y and w
are free because in the echelon form system they do not lead any row. They
are parameters because they are used in the solution set description. We could
have instead paramatrized with y and z by rewriting the second equation as
w =2/3− (1/3)z. In that case, the free variables are still y and w, but the
parameters are y and z. Notice that we could not have paramatrized with x and
y, so there is sometimes a restriction on the choice of parameters. The terms
‘parameter’ and ‘free’ are related because, as we shall show later in this chapter,
the solution set of a system can always be paramatrized with the free variables.
Consequenlty, we shall paramatrize all of our descriptions in this way.)
2.5 Example This is another system with infinitely many solutions.
x +2y =1
2x + z =2
3x +2y + z − w =4
−2ρ1+ρ2
−→
−3ρ1+ρ3
−ρ2+ρ3
−→
x + 2y =1
−4y + z =0
−4y + z − w =1
x + 2y =1
−4y + z =0
−w =1
The leading variables are x, y, andw. The variable z is free. (Notice here that,
although there are infinitely many solutions, the value of one of the variables is
fixed — w = −1.) Write w in terms of z with w = −1+0z. Then y =(1/4)z.
To express x in terms of z, substitute for y into the first equation to get x =
1 − (1/2)z. The solution set is {(1 − (1/2)z,(1/4)z,z,−1) � � z ∈ R}.
We finish this subsection by developing the notation for linear systems and
their solution sets that we shall use in the rest of this book.
2.6 Definition An m×n matrix is a rectangular array of numbers with m rows
and n columns. Each number in the matrix is an entry,