solve a system of linear equations by reducing a matrix pamela leutwyler
TRANSCRIPT
Solve a system of linear equations
By reducing a matrix
Pamela Leutwyler
0433
02
0322
zyxw
zyxw
zyxw
Describe the solutions to:
0433
02
0322
zyxw
zyxw
zyxw
Describe the solutions to:
01433
01211
01322
The coefficient matrix:
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
01322
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
01322
Replace row 1 with row 1 – row 2
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
01322
Replace row 1 with row 1 – row 2
-1
1 1 1 0
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
00111
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
00111
Replace row 2 with row 2 – row 1
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
00111
Replace row 2 with row 2 – row 1
-1
0433
02
0322
zyxw
zyxw
zyxw
01433
01211
00111
Replace row 2 with row 2 – row 1
-10 0 1 1
0433
02
0322
zyxw
zyxw
zyxw
01433
01100
00111
0433
02
0322
zyxw
zyxw
zyxw
01433
01100
00111
Replace row 3 with row 3 – 3 row 1
0433
02
0322
zyxw
zyxw
zyxw
01433
01100
00111
Replace row 3 with row 3 – 3 row 1
-3
0433
02
0322
zyxw
zyxw
zyxw
01433
01100
00111
Replace row 3 with row 3 – 3 row 1
-3
0 0 1 1
0433
02
0322
zyxw
zyxw
zyxw
01100
01100
00111
0433
02
0322
zyxw
zyxw
zyxw
01100
01100
00111
Replace row 3 with row 3 – row 2
0433
02
0322
zyxw
zyxw
zyxw
01100
01100
00111
Replace row 3 with row 3 – row 2
-1
0433
02
0322
zyxw
zyxw
zyxw
01100
01100
00111
Replace row 3 with row 3 – row 2
-1
0 0
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
00111
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
00111
Replace row 1 with row 1 – row 2
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
00111
Replace row 1 with row 1 – row 2
-1
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
00111
Replace row 1 with row 1 – row 2
-1
0 -1
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
01011
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
01011
The matrix is in REDUCED ECHELON FORM
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
01011
The matrix is in REDUCED ECHELON FORM
11
Every FNZE is a 1First NonZero Entry in a row of a matrix
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
01011
The matrix is in REDUCED ECHELON FORM
11
Every FNZE is a 1Every FNZE is to the right of the FNZE above it.
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
01011
The matrix is in REDUCED ECHELON FORM
11
Every FNZE is a 1Every FNZE is to the right of the FNZE above it.
00
0
0
In a column containing a FNZE, every other element is a 0
0433
02
0322
zyxw
zyxw
zyxw
00000
01100
01011
Interpret :
0433
02
0322
zyxw
zyxw
zyxw
Interpret :
00000
01100
01011
zyxw
zyxw
zyxw w= -x +z
0433
02
0322
zyxw
zyxw
zyxw
Interpret :
00000
01100
01011
zyxw
zyxw
zyxw w= -x +z
y= -z
0433
02
0322
zyxw
zyxw
zyxw
Interpret :
w= -x +z
y= -z
0433
02
0322
zyxw
zyxw
zyxw
Interpret :
w= -x +z
y= -zx= x
z= z
x and z are theindependent variables
0433
02
0322
zyxw
zyxw
zyxw
Interpret :
w= -x +z
y= -zx= x
z= z zz
zy
xx
zxw
1
1
1
11
0433
02
0322
zyxw
zyxw
zyxw
Interpret :
zz
zy
xx
zxw
1
1
1
11
1
1
0
1
0
0
1
1
zx
z
y
x
w
Every solution is a linear combination of these vectors