Section I. Solving Linear Systems
The solution set
25
0
1
1/3
B1C
B
C
cfw_B
C w | w 2 R
@1/3A
1
has many vectors besides the zero vector (if we interpret w as a number of
molecules then solutions make sense only when w is a nonnegative multiple of
3).
3.6
Chapter One. Linear Systems
14
2.4 Example The list of leading variables may skip over some columns. After
this reduction
2x - 2y
=0
z + 3w = 2
3x - 3y
=0
x - y + 2z + 6w = 4
2x - 2y
-(3/2)1 +3
!
-(1/2)1 +4
2x - 2y
-22 +4
!
=0
z + 3w = 2
0=0
2z + 6w = 4
=
16
Chapter One. Linear Systems
Matrix notation also claries the descriptions of solution sets. Example 2.3s
cfw_ (2 - 2z + 2w, -1 + z - w, z, w) | z, w 2 R is hard to read. We will rewrite it
to group all of the constants together, all of the coecients o
Section I. Solving Linear Systems
illustrates.
23
01
01
0
1
0
1
1/2
B4C
B-1C
B -1 C
BC
BC
B
C
BC
BC
B
C
cfw_ B0C + w B 3 C + u B1/2C | w, u 2 R
BC
BC
B
C
@0A
@1A
@0A
0
0
1
particular
solution
unrestricted
combination
The combination is unrestricted in th
Chapter One. Linear Systems
20
(a) a2,1
(b) a1,2
(c) a2,2
(d) a3,1
X 2.16 Give the size of each 0
matrix. 1
1
1
104
(a)
(b) @-1 1 A
215
3 -1
(c)
5
10
10
5
X 2.17 Do the indicated vector operation, if it is dened.
01 01
01 01
2
3
1
3
4
(a) @1A + @0A
(b) 5
Chapter One. Linear Systems
12
(a) 19
(b) 21
(c) 23
(d) 29
(e) 17
? X 1.37 [Am. Math. Mon., Jan. 1935] Laugh at this: AHAHA + TEHE = TEHAW. It
resulted from substituting a code letter for each digit of a simple example in
addition, and it is required to i
Chapter One. Linear Systems
10
is an equation with only one variable. From that we get the rst number in the
solution and then we get the rest with back-substitution. This method takes longer
than Gausss Method, since it involves more arithmetic operation
Chapter One
Linear Systems
I
Solving Linear Systems
Systems of linear equations are common in science and mathematics. These two
examples from high school science [Onan] give a sense of how they arise.
The rst example is from Statics. Suppose that we have
Section I. Solving Linear Systems
3
Finding the set of all solutions is solving the system. We dont need guesswork
or good luck; there is an algorithm that always works. This algorithm is Gausss
Method (or Gaussian elimination or linear elimination ).
1.4
Section I. Solving Linear Systems
5
1.6 Denition The three operations from Theorem 1.5 are the elementary reduction operations, or row operations , or Gaussian operations. They are
swapping , multiplying by a scalar (or rescaling ), and row combination .
Section I. Solving Linear Systems
7
Strictly speaking, to solve linear systems we dont need the row rescaling
operation. We have introduced it here because it is convenient and because we
will use it later in this chapter as part of a variation of Gausss