This document introduces systems of linear equations and matrix operations. It defines linear equations, solutions, and graphs related to systems of linear equations. It also defines matrices, matrix operations, and the relationship between systems of linear equations and matrices. Specifically, it describes how a system of linear equations can be represented by an augmented matrix and how elementary row operations on a matrix correspond to elementary transformations of the associated system of linear equations.
2. Ch1_2
Definition
• An equation such as x+3y=9 is called a linear equation.
( 線性方程式 )
• The graph of this equation is a straight line in the x-y plane.
• A pair of values of x and y that satisfy the equation is called
a solution.
1.1 Matrices and Systems of
Linear Equations
system of linear equations ( 線性聯立方程式 )
3. Ch1_3
Definition
A linear equation in n variables ( 變數 ) x1, x2, x3, …, xn has
the form
a1 x1 + a2 x2 + a3 x3 + … + an xn = b
where the coefficients ( 係數 ) a1, a2, a3, …, an and b are
real numbers ( 實數 ).
常見數系的英文名稱：
natural number ( 自然數 ), integer ( 整數 ), rational number ( 有理數 ),
real number ( 實數 ), complex number ( 複數 )
positive ( 正 ), negative ( 負 )
4. Ch1_4
Figure 1.2
No solution ( 無解 )
–2x + y = 3
–4x + 2y = 2
Lines are parallel.
No point of intersection.
No solutions.
Solutions for system of linear equations
Figure 1.1
Unique solution ( 唯一解 )
x + 3y = 9
–2x + y = –4
Lines intersect at (3, 2)
Unique solution:
x = 3, y = 2.
Figure 1.3
Many solution ( 無限多
解 )
4x – 2y = 6
6x – 3y = 9
Both equations have the
same graph. Any point on
the graph is a solution.
Many solutions.
5. Ch1_5
A linear equation in three variables corresponds to
a plane in three-dimensional ( 三維 ) space.
Unique solution
※ Systems of three linear equations in three variables:
7. Ch1_7
How to solve a system of linear equations?
Gauss-Jordan elimination. ( 高斯 - 喬登消去法 )
1.2 節會介紹
8. Ch1_8
Definition
• A matrix ( 矩陣 ) is a rectangular array of numbers.
• The numbers in the array are called the elements ( 元素 ) of
the matrix.
Matrices
−=
−
=
−
−
=
1298
520
653
C
38
50
17
B
157
432
A
注意矩陣左右兩邊是中括號不是直線，直線表示的是行列式。
11. Ch1_11
matrix of coefficient and augmented matrix
62
332
2
321
321
321
−=−−
=++
=++
xxx
xxx
xxx
Relations between system of linear equations
and matrices
係數矩陣 擴大矩陣
隨堂作業： 5(f)
tcoefficienofmatrix
211
132
111
−−
matrixaugmented
6211
3132
2111
−−−
12. Ch1_12
給定聯立方程式後，
不會改變解的一些轉換
Elementary Transformation
1. Interchange two equations.
2. Multiply both sides of an
equation by a nonzero
constant.
3. Add a multiple of one
equation to another equation.
將左邊的轉換對應到矩陣上
Elementary Row
Operation ( 基本列運算 )
1. Interchange two rows of a
matrix.
( 兩列交換 )
2. Multiply the elements of a
row by a nonzero constant.
( 某列的元素同乘一非零常數 )
3. Add a multiple of the
elements of one row to the
corresponding elements of
another row.
( 將一個列的倍數加進另一列裡 )
Elementary Row Operations of Matrices
17. Ch1_17
Summary
−−−
−
−
=
7012
32863
441284
]:[ BA
A BUse row operations to [A: B] :
.
1100
3010
2001
−
≈≈
−−−
−
−
7012
32863
441284
]:[]:[ XIBA n≈≈i.e.,
Def. [In : X] is called the reduced echelon form ( 簡化梯
式 )
of [A : B].Note. 1. If A is the matrix of coefficients of a system of n equations
in n variables that has a unique solution,
then A is row equivalent to In (A ≈ In).
2. If A ≈ In, then the system has unique solution.
72
32863
441284
21
321
321
−=−−
=−+
=−+
xx
xxx
xxx
18. Ch1_18
Example 4 Many Systems
Solving the following three systems of linear equation, all of
which have the same matrix of coefficients.
3321
2321
1321
42
for42
3
bxxx
bxxx
bxxx
=−+−
=+−
=+−
in turn
4
3
3
,
2
1
0
,
11
11
8
3
2
1
−
−
=
b
b
b
Solution
−−−−
−
−
4211421
3111412
308311
.
2
1
2
,
1
3
0
,
2
1
1
3
2
1
3
2
1
3
2
1
=
=
−=
=
=
=
=
−=
=
x
x
x
x
x
x
x
x
x
−−−
−−−
−
123110
315210
308311
−−−
212100
315210
013101
−
−
212100
131010
201001
R2+(–2)R1
R3+R1
≈
1)R2(R3
R2R1
−+
+
≈
R32R2
R3)1(R1
+
−+
≈
隨堂作業： 13(b)
The solutions to
the three systems are
20. Ch1_20
1-2 Gauss-Jordan Elimination
Definition
A matrix is in reduced echelon form ( 簡化梯式 ) if
1. Any rows consisting entirely of zeros are grouped at the
bottom of the matrix.
2. The first nonzero element of each other row is 1. This
element is called a leading 1.
3. The leading 1 of each row after the first is positioned to
the right of the leading 1 of the previous row.
4. All other elements in a column that contains a leading 1
are zero.
27. Ch1_27
Example 5
This example illustrates a system that has no solution. Let us try
to solve the system
122
32
4522
32
32
321
321
321
=+
−=−+
=+−
=+−
xx
xxx
xxx
xxx
Solution( 自行練習 )
≈
−
−−
−
↔
1220
2100
6330
3211
R3R2
( )
≈
−
−−
−
1220
2100
2110
3211
R2
3
1
≈
−
−−
−+
+
5400
2100
2110
1101
2)R2(R4
R2R1
≈
−
−
−+
+
−+
13000
2100
4010
3001
4)R3R(R4
R3R2
1)R3(R1
( )
≈
−
−
1000
2100
4010
3001
R4
13
1
The system has no solution.
≈
−−
−
−
−−
−
−
−+
−+
1220
6330
2100
3211
1220
3121
4522
3211
1)R1(R3
2)R1(R2
0x1+0x2+0x3=1
隨堂作業： 5(d)
28. Ch1_28
Homogeneous System of linear
Equations
Definition
A system of linear equations is said to be homogeneous ( 齊
次 ) if all the constant terms ( 等號右邊的常數項 ) are zeros.
Example:
=+−−
=−+
0632
052
321
321
xxx
xxx
Observe that is a solution.0,0,0 321 === xxx
Theorem 1.1
A system of homogeneous linear equations in n variables always
has the solution x1 = 0, x2 = 0. …, xn = 0. This solution is called
the trivial solution.
29. Ch1_29
Homogeneous System of linear
Equations
Theorem 1.2
A system of homogeneous linear equations that has more
variables than equations has many solutions.
Note. 除 trivial solution 外，可能還有其他解。
隨堂作業： 8(e)
−
≈≈
−−
−
0410
0301
0632
0521
=+−−
=−+
0632
052
321
321
xxx
xxx
The system has other nontrivial solutions.
rxrxrx ==−=∴ 321 ,4,3
Example:
