n unknowns x , x , , xn - math.uh.eduacaglar/math3321f17/ch5-slides-part1-sum... · n unknowns, x1,...
TRANSCRIPT
![Page 1: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/1.jpg)
Chapter 5. Linear Algebra
A linear (algebraic) equation in
n unknowns, x1, x2, . . . , xn, is
an equation of the form
a1x1 + a2x2 + · · · + anxn = b
where a1, a2, . . . , an and b are
real numbers.
1
![Page 2: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/2.jpg)
linear equation in one unknown
x: ax = b.
2
![Page 3: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/3.jpg)
Exactly one of following holds:
(1) there is precisely one solution
x = b/a, a 6= 0,
(2) there are no solutions
0x = b, b 6= 0
(3) there are infinitely many solu-
tions
0x = 0.
3
![Page 4: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/4.jpg)
linear equation in two unknowns
x, y:
ax + by = α.
A solution of the equation is an or-
dered pair of numbers (x, y).
Assuming a, b, not both 0, then
the set of all ordered pairs that sat-
isfy the equation is a straight line
(in the x, y-plane).
4
![Page 5: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/5.jpg)
A system of two linear equations in
two unknowns:
ax + by = α
cx + dy = β
Find ordered pairs (x, y) that sat-
isfy both equations simultaneously.
5
![Page 6: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/6.jpg)
Two lines in the plane either
(a) have a unique point of intersec-
tion (the lines have different slopes),
that is, the system has exactly one
solution
-4 -3 -2 -1 1 2 3
-2
2
4
6
8
10
12
6
![Page 7: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/7.jpg)
(b) are parallel (the lines have the
same slope but, for example, differ-
ent y-intercepts)
that is, the system has no solution
-3 -2 -1 1 2 3
1
2
3
7
![Page 8: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/8.jpg)
(c) coincide (same slope, same y-
intercept),
that is, the system has infinitely many
solutions.
-3 -2 -1 1 2 3
1
2
8
![Page 9: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/9.jpg)
That is, there is either a
(a) unique solution,
(b) no solution,
or
(c) infinitely many solutions.
9
![Page 10: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/10.jpg)
A system of three linear equations
in two unknowns:
ax + by = α
cx + dy = β
ex + fy = γ
Find ordered pairs (x, y) that sat-
isfy the three equations simultane-
ously.
10
![Page 11: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/11.jpg)
There is either a
(a) unique solution,
(b) no solution, (this is usually
what happens)
or
(c) infinitely many solutions.
11
![Page 12: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/12.jpg)
Example:
x + y = 2−2x + y = 24x + y = 11
-1 1 2 3
5
10
12
![Page 13: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/13.jpg)
A linear equation in three unknowns
x, y, z:
ax + by + cz = α.
A solution of the equation is an or-
dered triple of numbers (x, y, z).
13
![Page 14: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/14.jpg)
Assuming a, b, c, not all 0, then
the set of all ordered triples that
satisfy the equation is a plane (in
3-space).
-5 0 5
-5
0
5
-20
0
20
14
![Page 15: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/15.jpg)
A system of two linear equations in
three unknowns
a11x + a12y + a13z = b1
a21x + a22y + a23z = b2
• Either the two planes are parallel
(the system has no solutions),
• they coincide (infinitely many so-
lutions, a whole plane of solutions),
15
![Page 16: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/16.jpg)
• they intersect in a straight line
(again, infinitely many solutions.)
-5 0 5
-5
0
5
-20
0
20
16
![Page 17: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/17.jpg)
A system of three linear equations
in three unknowns.
a11x + a12y + a13z = b1
a21x + a22y + a23z = b2
a31x + a32y + a33z = b3
The system represents three planes
in 3-space.
17
![Page 18: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/18.jpg)
(a) The system has a unique solu-
tion; the three planes have a unique
point of intersection;
(b) The system has infinitely many
solutions; the three planes inter-
sect in a line, or the three planes
intersect in a plane.
(c) The system has no solution; there
is no point the lies on all three
planes.18
![Page 19: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/19.jpg)
Systems of Linear Algebraic Equa-
tions
Example 1: Solve the system
x + 3y = −5
2x − y = 4
19
![Page 20: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/20.jpg)
Equivalent system
x + 3y = −5
y = −2
Solution set:
x = 1, y = −2
20
![Page 21: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/21.jpg)
Example 2: Solve the system
x + 2y − 5z = −1−3x − 9y + 21z = 0
x + 6y − 11z = 1
21
![Page 22: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/22.jpg)
Example 2 con’t
22
![Page 23: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/23.jpg)
Equivalent system
x + 2y − 5z = −1y − 2z = 1
z = −1
Solution set:
x = −4, y = −1, z = −1
23
![Page 24: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/24.jpg)
Example 3: Solve the system
3x − 4y − z = 32x − 3y + z = 1x − 2y + 3z = 2
24
![Page 25: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/25.jpg)
Example 3 con’t
25
![Page 26: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/26.jpg)
Equivalent system
x − 2y + 3z = 2y − 5z = −3
0z = 1
The system has no solution.
26
![Page 27: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/27.jpg)
The Elementary Operations
The operations that produce equiv-
alent systems are called elementary
operations.
1. Multiply an equation by a nonzero
number.
2. Interchange two equations.
3. Multiply an equation by a num-
ber and add it to another equation.
27
![Page 28: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/28.jpg)
Example 4: Solve the system
x1 − 2x2 + x3 − x4 = −2
−2x1 + 5x2 − x3 + 4x4 = 1
3x1 − 7x2 + 4x3 + 3x4 = −5
28
![Page 29: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/29.jpg)
Example 4 con’t
29
![Page 30: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/30.jpg)
Equivalent system
x1 − 2x2 + x3 − x4 = −2
x2 + x3 + 2x4 = −3
x3 + 4x4 = −1
30
![Page 31: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/31.jpg)
Solution set:
x1 = −5 + 17a,
x2 = −2 + 6a,
x3 = −1 − 4a,
x4 = a, a any real number.
31
![Page 32: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/32.jpg)
Terms
A matrix is a rectangular array of
numbers. A matrix with m rows and
n columns is an m × n matrix.
Matrix representation of a sys-
tem of linear equations
a11x1 + a12x2 + · · · + a1nxn = b1a21x1 + a22x2 + · · · + a2nxn = b2
... ... ... ... ...
am1x1 + am2x2 + · · · + amnxn = bm
32
![Page 33: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/33.jpg)
Augmented matrix and matrix of
coefficients:
Augmented matrix:
a11 a12 · · · a1n b1a21 a22 · · · a2n b2... ... ... ... ...
am1 am2 · · · amn bm
Matrix of coefficients:
a11 a12 · · · a1na21 a22 · · · a2n... ... ...
am1 a32 · · · amn
33
![Page 34: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/34.jpg)
Elementary row operations:
1. Interchange row i and row j
Ri ↔ Rj.
2. Multiply row i by a nonzero
number k
kRi → Ri.
3. Multiply row i by a number k
and add the result to row j
kRi + Rj → Rj.
34
![Page 35: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/35.jpg)
Examples
1. Solve the system
x + 2y − 5z = −1−3x − 9y + 21z = 0
x + 6y − 11z = 1
Augmented matrix:
1 2 −5 −1−3 −9 21 01 6 −11 1
35
![Page 36: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/36.jpg)
Row reduce
1 2 −5 −1−3 −9 21 01 6 −11 1
36
![Page 37: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/37.jpg)
1 2 −5 −10 1 −2 10 0 1 −1
Corresponding (equivalent) system
of equations:
x + 2y − 5z = −1
y − 2z = 1
z = −1
Solution set:
x = −4, y = −1, z = −1
37
![Page 38: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/38.jpg)
2. Solve the system:
3x − 4y − z = 32x − 3y + z = 1x − 2y + 3z = 2
Augmented matrix:
3 −4 −1 32 −3 1 11 −2 3 2
.
38
![Page 39: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/39.jpg)
Row reduce
3 −4 −1 32 −3 1 11 −2 3 2
39
![Page 40: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/40.jpg)
Equivalent system
1 −2 3 20 1 −5 −30 0 0 1
Corresponding system of equations:
x − 2y + 3z = 20x + y − 5z = −3
0x + 0y + 0z = 1
That is
x − 2y + 3z = 2y − 5z = −3
0z = 1
Solution set: no solution.
40
![Page 41: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/41.jpg)
3. Solve the system
x + y − 3z = 12x + y − 4z = 0
−3x + 2y − z = 7
Augmented matrix:
1 1 −3 12 1 −4 0
−3 2 −1 7
41
![Page 42: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/42.jpg)
Row reduce
1 1 −3 12 1 −4 0
−3 2 −1 7
42
![Page 43: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/43.jpg)
Equivalent system:
1 1 −3 10 1 −2 20 0 0 0
.
Corresponding system of equations:
x + y − 3z = 10x + y − 2z = 2
0x + 0y + 0z = 0
or
x + y − 3z = 1y − 2z = 2
0z = 0
or
x + y − 3z = 1y − 2z = 2
43
![Page 44: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/44.jpg)
This system has infinitely many so-
lutions given by:
x = −1 + a,
y = 2 + 2a,
z = a, a any real number.
44
![Page 45: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/45.jpg)
Row echelon form:
1. Rows consisting entirely of ze-
ros are at the bottom of the matrix.
2. The first nonzero entry in a
nonzero row is a 1. This is called
the leading 1.
3. If row i and row i + 1 are
nonzero rows, then the leading 1 in
row i+1 is to the right of the leading
1 in row i.45
![Page 46: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/46.jpg)
1 2 −5 −10 1 −2 10 0 1 −1
(Example 1)
1 −2 3 20 1 −5 −30 0 0 1
(Example 2)
1 1 −3 10 1 −2 20 0 0 0
(Example 3)
46
![Page 47: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/47.jpg)
NOTE:
1. All the entries below a leading
1 are zero.
2. The number of leading 1’s is
less than or equal to the number of
rows.
3. The number of leading 1’s is
less than or equal to the number of
columns.
47
![Page 48: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/48.jpg)
Solution method for systems of
linear equations:
1. Write the augmented matrix
(A|b) for the system.
2. Use elementary row operations
to transform the augmented matrix
to row echelon form.
3. Write the system of equa-
tions corresponding to the row ech-
elon form.48
![Page 49: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/49.jpg)
4. Back substitute to find the
solution set.
This method is called Gaussian elim-
ination with back substitution.
49
![Page 50: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/50.jpg)
Consistent/Inconsistent systems:
A system of linear equations is con-
sistent if it has at least one solu-
tion.
That is, a system is consistent if it
has either a unique solution or in-
finitely many solutions.
A system that has no solutions is
inconsistent.
50
![Page 51: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/51.jpg)
Consistent systems:
A consistent system is said to be
independent if it has a unique so-
lution.
A system with infinitely many solu-
tions is called dependent.
51
![Page 52: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/52.jpg)
4. Solve the system of equations
2x1 + 5x2 − 5x3 − 7x4 = 8x1 + 2x2 − 3x3 − 4x4 = 2
−3x1 − 6x2 + 11x3 + 16x4 = 0
Augmented matrix:
2 5 −5 −7 81 2 −3 −4 2
−3 −6 11 16 0
.
52
![Page 53: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/53.jpg)
Transform to row echelon form:
2 5 −5 −7 81 2 −3 −4 2
−3 −6 11 16 0
.
53
![Page 54: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/54.jpg)
Equivalent system:
1 2 −3 −4 20 1 1 1 40 0 1 2 3
.
Corresponding system of equations:
x1 + 2x2 − 3x3 − 4x4 = 2
x2 + x3 + x4 = 4
x3 + 2x4 = 3
54
![Page 55: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/55.jpg)
Solution set:
x1 = 9 − 4a,
x2 = 1 + a,
x3 = 3 − 2a,
x4 = a, a any real number.
55
![Page 56: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/56.jpg)
5. Solve the system of equations
x1 − 3x2 + 2x3 − x4 + 2x5 = 2
3x1 − 9x2 + 7x3 − x4 + 3x5 = 7
2x1 − 6x2 + 7x3 + 4x4 − 5x5 = 7
Augmented matrix:
1 −3 2 −1 2 23 −9 7 −1 3 72 −6 7 4 −5 7
56
![Page 57: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/57.jpg)
Transform to row echelon form:
1 −3 2 −1 2 23 −9 7 −1 3 72 −6 7 4 −5 7
57
![Page 58: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/58.jpg)
Equivalent system:
1 −3 2 −1 2 20 0 1 2 −3 10 0 0 0 0 0
.
Corresponding system of equations:
x1 − 3x2 + 2x3 − x4 + 2x5 = 20x1 + 0x2 + x3 + 2x4 − 3x5 = 1
0x1 + 0x2 + 0x3 + 0x4 + 0x5 = 0.
which is
x1 − 3x2 + 2x3 − x4 + 2x5 = 2x3 + 2x4 − 3x5 = 1
58
![Page 59: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/59.jpg)
Solution set:
x1 = 3a + 5b − 8c,
x2 = a,
x3 = 1 − 2b + 3c,
x4 = b,
x5 = c,
a, b, c arbitrary real numbers
59
![Page 60: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/60.jpg)
6. For what value(s) of k, if any,
does the system
x + y − z = 1
2x + 3y + kz = 3
x + ky + 3z = 2
have:
(a) a unique solution?
(b) infinitely many solutions?
(c) no solution?
60
![Page 61: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/61.jpg)
1 1 −1 12 3 k 31 k 3 2
61
![Page 62: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/62.jpg)
1 1 −1 10 1 k + 2 10 0 (k + 3)(k − 2) k − 2
(a) Unique solution: k 6= 2,−3.
(b) Infinitely many solns: k = 2.
(c) No solution: k = −3.
62
![Page 63: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/63.jpg)
If an m × n matrix A is reduced to
row echelon form, then the number
of non-zero rows in its row echelon
form is called the rank of A.
Equivalently, the rank of a matrix is
the number of leading 1’s in its row
echelon form.
63
![Page 64: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/64.jpg)
Note:
1. The rank of a matrix is less
than or equal to the number of rows.
(Obvious)
2. The rank of a matrix is also
less than or equal to the number of
columns.
64
![Page 65: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/65.jpg)
Consistent/Inconsistent Systems
Case 1: If the last nonzero row in
the row echelon form of the aug-
mented matrix is
(0 0 0 · · · 0 |1),
then that row corresponds to the
equation
0x1 + 0x2 + 0x3 + · · · + 0xn = 1,
which has no solutions. Therefore,
the system has no solutions.65
![Page 66: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/66.jpg)
Note: In this case, rank of aug-
mented matrix > rank of coefficient
matrix
![Page 67: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/67.jpg)
Case 2: If the last nonzero row has
the form
(0 0 0 · · · 1 ∗ · · · ∗ | b),
where the “1” is in the kth, k ≤ n
column, then the row corresponds
to the equation
0x1+· · ·+0xk−1+xk+(∗)xk+1+· · ·+(∗)xn = b
and the system either has a unique
solution or infinitely many solutions.
66
![Page 68: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/68.jpg)
NOTE: In this case, rank of aug-
mented matrix = the rank of coef-
ficient matrix
67
![Page 69: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/69.jpg)
Theorem: A system of linear equa-
tions is consistent if and only if the
rank of the coefficient matrix equals
the rank of the augmented matrix.
68
![Page 70: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/70.jpg)
5.4. Reduced Row Echelon Form
Example Solve the system (c.f. Ex-
ample 1)
x + 2y − 5z = −1
−3x − 9y + 21z = 0
x + 6y − 11z = 1
Augmented matrix:
1 2 −5 −1−3 −9 21 01 6 −11 1
Row reduce to:
1 2 −5 −10 1 −2 10 0 1 −1
69
![Page 71: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/71.jpg)
Corresponding (equivalent) system
of equations
x + 2y − 5z = −1
y − 2z = 1
z = −1
Back substitute to get:
x = −4, y = −1, z = −1.
70
![Page 72: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/72.jpg)
Or, continue row operations:
1 2 −5 −10 1 −2 10 0 1 −1
→
Corresponding system of equations
x = −4
y = −1
z = −1
71
![Page 73: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/73.jpg)
2. Solve the system (c.f. Example
4)
2x1 + 5x2 − 5x3 − 7x4 = 7
x1 + 2x2 − 3x3 − 4x4 = 2
−3x1 − 6x2 + 11x3 + 16x4 = 0
Augmented matrix:
2 5 −5 −7 71 2 −3 −4 2
−3 −6 11 16 0
.
Row echelon form:
1 2 −3 −4 20 1 1 1 30 0 1 2 3
.
72
![Page 74: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/74.jpg)
Corresponding system of equations:
x1 + 2x2 − 3x3 − 4x4 = 2
x2 + x3 + x4 = 3
x3 + 2x4 = 3
Solution set:
x1 = 11 − 4a,
x2 = a,
x3 = 3 − 2a,
x4 = a, a any real number.
73
![Page 75: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/75.jpg)
Alternative solution: Continue the
row operations
1 2 −3 −4 20 1 1 1 30 0 1 2 3
→
74
![Page 76: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/76.jpg)
1 0 0 4 110 1 0 −1 00 0 1 2 3
Corresponding system of equations:
x1 + 4x4 = 11
x2 − x4 = 0
x3 + 2x4 = 3
x1 = 11−4a, x2 = a, x3 = 3−2a, x4 = a,
a any real number.
75
![Page 77: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/77.jpg)
3. Solve the system of equations
2x1 +3x2 −5x3 −2x4 = 2−2x1 −4x2 +4x3 −3x4 = 6
x1 +2x2 −2x3 +3x4 = 0
76
![Page 78: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/78.jpg)
Reduced Row Echelon Form
1. Rows consisting entirely of zeros
are at the bottom of the matrix.
2. The first nonzero entry in a
nonzero row is a 1.
3. The leading 1 in row i + 1 is to
the right of the leading 1 in row i.
4. The leading 1 is the only nonzero
entry in its column.
77
![Page 79: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/79.jpg)
Homogeneous Systems
The system of linear equations
a11x1 + a12x2 + · · · + a1nxn = b1a21x1 + a22x2 + · · · + a2nxn = b2
... = ...am1x1 + am2x2 + · · · + amnxn = bm
is homogeneous if
b1 = b2 = · · · = bm = 0,
otherwise, the system is nonhomo-
geneous.
C.f. Linear differential equations.78
![Page 80: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/80.jpg)
A homogeneous system
a11x1 + a12x2 + · · · + a1nxn = 0
a21x1 + a22x2 + · · · + a2nxn = 0... ... ... ... ...
am1x1 + am2x2 + · · · + amnxn = 0
ALWAYS has at least one solution,
namely
x1 = x2 = · · · = xn = 0,
called the trivial solution
That is, homogeneous systems are
always CONSISTENT.
79
![Page 81: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/81.jpg)
3. Solve the homogeneous system
x − 2y + 2z = 0
4x − 7y + 3z = 0
2x − y + 2z = 0
Augmented matrix:
1 −2 2 04 −7 3 02 −1 2 0
Row echelon form:
1 −2 2 00 1 −5 00 0 1 0
80
![Page 82: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/82.jpg)
Corresponding system of equations:
x − 2y + 2z = 0
y − 5z = 0
z = 0.
This system has the unique solution
x = 0,
y = 0,
z = 0.
The trivial solution is the only solu-
tion.
81
![Page 83: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/83.jpg)
4. Solve the homogeneous system
3x − 2y + z = 0
x + 4y + 2z = 0
7x + 4z = 0
Augmented matrix:
3 −2 1 01 4 2 07 0 4 0
Row echelon form:
1 4 2 00 1 5/14 00 0 0 0
82
![Page 84: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/84.jpg)
Corresponding system of equations:
x + 4y + 2z = 0
y +5
14z = 0
This system has infinitely many so-
lutions:
x = −2
7a,
y = −5
14a,
z = a, a any real number.
83
![Page 85: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/85.jpg)
5.
2x1 + 5x2 − 5x3 + 7x4 + 5x5 = 2
x1 + 2x2 − 3x3 + 4x4 + 2x5 = 3
−3x1 − 6x2 + 9x3 − 10x4 − 2x5 = 0
Augmented matrix:
1 2 −3 4 2 32 5 −5 7 5 2
−3 −6 9 −10 −2 −3
.
84
![Page 86: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/86.jpg)
1 2 −3 4 2 32 5 −5 7 5 2
−3 −6 9 −10 −2 −3
.
85
![Page 87: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/87.jpg)
Reduced row echelon form:
1 0 −5 0 −12 −70 1 1 −1 1 −40 0 0 1 2 3
86
![Page 88: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/88.jpg)
Corresponding system of equations:
x1 − 5x3 − 12x5 = −7
x2 + x3 + 3x5 = −1
x4 + 2x5 = 3
Solution set:
x1 = −7 + 5a + 12b,
x2 = −1 − a − 3b
x3 = a
x4 = 3 − 2b,
x5 = b a, b arbitrary real nos.
87
![Page 89: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/89.jpg)
Write as an ordered quintuple:
x1x2x3x4x5
=
−7 + 5a + 12b−1 − a − 3b
a3 − 2b
b
= a
5−1100
+ b
12−30
−21
+
−7−1030
88
![Page 90: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/90.jpg)
What can you say about
−7−1030
, and
5−1100
,
12−30
−21
?
89
![Page 91: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/91.jpg)
a
5−1100
+ b
12−30
−21
is the set of all solutions of the ho-
mogeneous system
2x1 + 5x2 − 5x3 + 7x4 + 5x5 = 0
x1 + 2x2 − 3x3 + 4x4 + 2x5 = 0
−3x1 − 6x2 + 9x3 − 10x4 − 2x5 = 0
90
![Page 92: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/92.jpg)
−7−1030
is a solution of the given
nonhomogeneous system
2x1 + 5x2 − 5x3 + 7x4 + 5x5 = 2
x1 + 2x2 − 3x3 + 4x4 + 2x5 = 3
−3x1 − 6x2 + 9x3 − 10x4 − 2x5 = 0
91
![Page 93: n unknowns x , x , , xn - math.uh.eduacaglar/MATH3321F17/Ch5-slides-Part1-Sum... · n unknowns, x1, x2, ..., xn, is an equation of the form a1x1 + a2x2 + ··· + anxn = b where a1,](https://reader030.vdocuments.mx/reader030/viewer/2022021610/5bf42b4009d3f2be5a8c1d77/html5/thumbnails/93.jpg)
That is, the set of solutions of the
nonhomogeneous system consists of
• the set of all solutions of the cor-
responding homogeneous system
plus
• one solution of the nonhomoge-
neous system.
C.f. Section 3.4.
92