Algebra Examples
S([abc])=[a+3b-6c2a+b+ca+5b+c]S⎛⎜⎝⎡⎢⎣abc⎤⎥⎦⎞⎟⎠=⎡⎢⎣a+3b−6c2a+b+ca+5b+c⎤⎥⎦
Step 1
The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation).
[a+3b-6c2a+b+ca+5b+c]=0⎡⎢⎣a+3b−6c2a+b+ca+5b+c⎤⎥⎦=0
Step 2
Create a system of equations from the vector equation.
a+3b-6c=0a+3b−6c=0
2a+b+c=02a+b+c=0
a+5b+c=0a+5b+c=0
Step 3
Write the system as a matrix.
[13-6021101510]⎡⎢
⎢⎣13−6021101510⎤⎥
⎥⎦
Step 4
Step 4.1
Perform the row operation R2=R2-2R1R2=R2−2R1 to make the entry at 2,12,1 a 00.
Step 4.1.1
Perform the row operation R2=R2-2R1R2=R2−2R1 to make the entry at 2,12,1 a 00.
[13-602-2⋅11-2⋅31-2⋅-60-2⋅01510]⎡⎢
⎢⎣13−602−2⋅11−2⋅31−2⋅−60−2⋅01510⎤⎥
⎥⎦
Step 4.1.2
Simplify R2R2.
[13-600-51301510]⎡⎢
⎢⎣13−600−51301510⎤⎥
⎥⎦
[13-600-51301510]⎡⎢
⎢⎣13−600−51301510⎤⎥
⎥⎦
Step 4.2
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Step 4.2.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[13-600-51301-15-31+60-0]⎡⎢
⎢⎣13−600−51301−15−31+60−0⎤⎥
⎥⎦
Step 4.2.2
Simplify R3R3.
[13-600-51300270]⎡⎢
⎢⎣13−600−51300270⎤⎥
⎥⎦
[13-600-51300270]⎡⎢
⎢⎣13−600−51300270⎤⎥
⎥⎦
Step 4.3
Multiply each element of R2R2 by -15−15 to make the entry at 2,22,2 a 11.
Step 4.3.1
Multiply each element of R2R2 by -15−15 to make the entry at 2,22,2 a 11.
[13-60-15⋅0-15⋅-5-15⋅13-15⋅00270]⎡⎢
⎢⎣13−60−15⋅0−15⋅−5−15⋅13−15⋅00270⎤⎥
⎥⎦
Step 4.3.2
Simplify R2R2.
[13-6001-13500270]⎡⎢
⎢⎣13−6001−13500270⎤⎥
⎥⎦
[13-6001-13500270]⎡⎢
⎢⎣13−6001−13500270⎤⎥
⎥⎦
Step 4.4
Perform the row operation R3=R3-2R2R3=R3−2R2 to make the entry at 3,23,2 a 00.
Step 4.4.1
Perform the row operation R3=R3-2R2R3=R3−2R2 to make the entry at 3,23,2 a 00.
[13-6001-13500-2⋅02-2⋅17-2(-135)0-2⋅0]⎡⎢
⎢
⎢
⎢⎣13−6001−13500−2⋅02−2⋅17−2(−135)0−2⋅0⎤⎥
⎥
⎥
⎥⎦
Step 4.4.2
Simplify R3R3.
[13-6001-1350006150]⎡⎢
⎢
⎢⎣13−6001−1350006150⎤⎥
⎥
⎥⎦
[13-6001-1350006150]⎡⎢
⎢
⎢⎣13−6001−1350006150⎤⎥
⎥
⎥⎦
Step 4.5
Multiply each element of R3R3 by 561561 to make the entry at 3,33,3 a 11.
Step 4.5.1
Multiply each element of R3R3 by 561561 to make the entry at 3,33,3 a 11.
[13-6001-1350561⋅0561⋅0561⋅615561⋅0]⎡⎢
⎢
⎢⎣13−6001−1350561⋅0561⋅0561⋅615561⋅0⎤⎥
⎥
⎥⎦
Step 4.5.2
Simplify R3R3.
[13-6001-13500010]⎡⎢
⎢⎣13−6001−13500010⎤⎥
⎥⎦
[13-6001-13500010]⎡⎢
⎢⎣13−6001−13500010⎤⎥
⎥⎦
Step 4.6
Perform the row operation R2=R2+135R3R2=R2+135R3 to make the entry at 2,32,3 a 00.
Step 4.6.1
Perform the row operation R2=R2+135R3R2=R2+135R3 to make the entry at 2,32,3 a 00.
[13-600+135⋅01+135⋅0-135+135⋅10+135⋅00010]⎡⎢
⎢⎣13−600+135⋅01+135⋅0−135+135⋅10+135⋅00010⎤⎥
⎥⎦
Step 4.6.2
Simplify R2R2.
[13-6001000010]⎡⎢
⎢⎣13−6001000010⎤⎥
⎥⎦
[13-6001000010]⎡⎢
⎢⎣13−6001000010⎤⎥
⎥⎦
Step 4.7
Perform the row operation R1=R1+6R3R1=R1+6R3 to make the entry at 1,31,3 a 00.
Step 4.7.1
Perform the row operation R1=R1+6R3R1=R1+6R3 to make the entry at 1,31,3 a 00.
[1+6⋅03+6⋅0-6+6⋅10+6⋅001000010]⎡⎢
⎢⎣1+6⋅03+6⋅0−6+6⋅10+6⋅001000010⎤⎥
⎥⎦
Step 4.7.2
Simplify R1.
[130001000010]
[130001000010]
Step 4.8
Perform the row operation R1=R1-3R2 to make the entry at 1,2 a 0.
Step 4.8.1
Perform the row operation R1=R1-3R2 to make the entry at 1,2 a 0.
[1-3⋅03-3⋅10-3⋅00-3⋅001000010]
Step 4.8.2
Simplify R1.
[100001000010]
[100001000010]
[100001000010]
Step 5
Use the result matrix to declare the final solution to the system of equations.
a=0
b=0
c=0
Step 6
Write a solution vector by solving in terms of the free variables in each row.
[abc]=[000]
Step 7
Write as a solution set.
{[000]}
Step 8
The kernel of S is the subspace {[000]}.
K(S)={[000]}
