Linear Algebra Examples
[-36-11-71-223-12-458-4]⎡⎢⎣−36−11−71−223−12−458−4⎤⎥⎦
Step 1
Write as an augmented matrix for Ax=0Ax=0.
[-36-11-701-223-102-458-40]⎡⎢
⎢⎣−36−11−701−223−102−458−40⎤⎥
⎥⎦
Step 2
Step 2.1
Multiply each element of R1R1 by -13−13 to make the entry at 1,11,1 a 11.
Step 2.1.1
Multiply each element of R1R1 by -13−13 to make the entry at 1,11,1 a 11.
[-13⋅-3-13⋅6-13⋅-1-13⋅1-13⋅-7-13⋅01-223-102-458-40]⎡⎢
⎢⎣−13⋅−3−13⋅6−13⋅−1−13⋅1−13⋅−7−13⋅01−223−102−458−40⎤⎥
⎥⎦
Step 2.1.2
Simplify R1R1.
[1-213-137301-223-102-458-40]⎡⎢
⎢⎣1−213−137301−223−102−458−40⎤⎥
⎥⎦
[1-213-137301-223-102-458-40]⎡⎢
⎢⎣1−213−137301−223−102−458−40⎤⎥
⎥⎦
Step 2.2
Perform the row operation R2=R2-R1R2=R2−R1 to make the entry at 2,12,1 a 00.
Step 2.2.1
Perform the row operation R2=R2-R1R2=R2−R1 to make the entry at 2,12,1 a 00.
[1-213-137301-1-2+22-133+13-1-730-02-458-40]⎡⎢
⎢
⎢⎣1−213−137301−1−2+22−133+13−1−730−02−458−40⎤⎥
⎥
⎥⎦
Step 2.2.2
Simplify R2R2.
[1-213-137300053103-10302-458-40]⎡⎢
⎢
⎢⎣1−213−137300053103−10302−458−40⎤⎥
⎥
⎥⎦
[1-213-137300053103-10302-458-40]⎡⎢
⎢
⎢⎣1−213−137300053103−10302−458−40⎤⎥
⎥
⎥⎦
Step 2.3
Perform the row operation R3=R3-2R1R3=R3−2R1 to make the entry at 3,13,1 a 00.
Step 2.3.1
Perform the row operation R3=R3-2R1R3=R3−2R1 to make the entry at 3,13,1 a 00.
[1-213-137300053103-10302-2⋅1-4-2⋅-25-2(13)8-2(-13)-4-2(73)0-2⋅0]⎡⎢
⎢
⎢
⎢
⎢⎣1−213−137300053103−10302−2⋅1−4−2⋅−25−2(13)8−2(−13)−4−2(73)0−2⋅0⎤⎥
⎥
⎥
⎥
⎥⎦
Step 2.3.2
Simplify R3R3.
[1-213-137300053103-103000133263-2630]⎡⎢
⎢
⎢
⎢⎣1−213−137300053103−103000133263−2630⎤⎥
⎥
⎥
⎥⎦
[1-213-137300053103-103000133263-2630]⎡⎢
⎢
⎢
⎢⎣1−213−137300053103−103000133263−2630⎤⎥
⎥
⎥
⎥⎦
Step 2.4
Multiply each element of R2R2 by 3535 to make the entry at 2,32,3 a 11.
Step 2.4.1
Multiply each element of R2R2 by 3535 to make the entry at 2,32,3 a 11.
[1-213-1373035⋅035⋅035⋅5335⋅10335(-103)35⋅000133263-2630]⎡⎢
⎢
⎢
⎢
⎢⎣1−213−1373035⋅035⋅035⋅5335⋅10335(−103)35⋅000133263−2630⎤⎥
⎥
⎥
⎥
⎥⎦
Step 2.4.2
Simplify R2R2.
[1-213-137300012-2000133263-2630]⎡⎢
⎢
⎢⎣1−213−137300012−2000133263−2630⎤⎥
⎥
⎥⎦
[1-213-137300012-2000133263-2630]⎡⎢
⎢
⎢⎣1−213−137300012−2000133263−2630⎤⎥
⎥
⎥⎦
Step 2.5
Perform the row operation R3=R3-133R2R3=R3−133R2 to make the entry at 3,33,3 a 00.
Step 2.5.1
Perform the row operation R3=R3-133R2R3=R3−133R2 to make the entry at 3,33,3 a 00.
[1-213-137300012-200-133⋅00-133⋅0133-133⋅1263-133⋅2-263-133⋅-20-133⋅0]⎡⎢
⎢
⎢⎣1−213−137300012−200−133⋅00−133⋅0133−133⋅1263−133⋅2−263−133⋅−20−133⋅0⎤⎥
⎥
⎥⎦
Step 2.5.2
Simplify R3R3.
[1-213-137300012-20000000]⎡⎢
⎢⎣1−213−137300012−20000000⎤⎥
⎥⎦
[1-213-137300012-20000000]⎡⎢
⎢⎣1−213−137300012−20000000⎤⎥
⎥⎦
Step 2.6
Perform the row operation R1=R1-13R2R1=R1−13R2 to make the entry at 1,31,3 a 00.
Step 2.6.1
Perform the row operation R1=R1-13R2R1=R1−13R2 to make the entry at 1,31,3 a 00.
[1-13⋅0-2-13⋅013-13⋅1-13-13⋅273-13⋅-20-13⋅00012-20000000]⎡⎢
⎢⎣1−13⋅0−2−13⋅013−13⋅1−13−13⋅273−13⋅−20−13⋅00012−20000000⎤⎥
⎥⎦
Step 2.6.2
Simplify R1R1.
[1-20-1300012-20000000]⎡⎢
⎢⎣1−20−1300012−20000000⎤⎥
⎥⎦
[1-20-1300012-20000000]⎡⎢
⎢⎣1−20−1300012−20000000⎤⎥
⎥⎦
[1-20-1300012-20000000]⎡⎢
⎢⎣1−20−1300012−20000000⎤⎥
⎥⎦
Step 3
Use the result matrix to declare the final solution to the system of equations.
x1-2x2-x4+3x5=0x1−2x2−x4+3x5=0
x3+2x4-2x5=0x3+2x4−2x5=0
0=00=0
Step 4
Write a solution vector by solving in terms of the free variables in each row.
[x1x2x3x4x5]=[2x2+x4-3x5x2-2x4+2x5x4x5]⎡⎢
⎢
⎢
⎢
⎢
⎢⎣x1x2x3x4x5⎤⎥
⎥
⎥
⎥
⎥
⎥⎦=⎡⎢
⎢
⎢
⎢
⎢
⎢⎣2x2+x4−3x5x2−2x4+2x5x4x5⎤⎥
⎥
⎥
⎥
⎥
⎥⎦
Step 5
Write the solution as a linear combination of vectors.
[x1x2x3x4x5]=x2[21000]+x4[10-210]+x5[-30201]⎡⎢
⎢
⎢
⎢
⎢
⎢⎣x1x2x3x4x5⎤⎥
⎥
⎥
⎥
⎥
⎥⎦=x2⎡⎢
⎢
⎢
⎢
⎢
⎢⎣21000⎤⎥
⎥
⎥
⎥
⎥
⎥⎦+x4⎡⎢
⎢
⎢
⎢
⎢
⎢⎣10−210⎤⎥
⎥
⎥
⎥
⎥
⎥⎦+x5⎡⎢
⎢
⎢
⎢
⎢
⎢⎣−30201⎤⎥
⎥
⎥
⎥
⎥
⎥⎦
Step 6
Write as a solution set.
{x2[21000]+x4[10-210]+x5[-30201]|x2,x4,x5∈R}
Step 7
The solution is the set of vectors created from the free variables of the system.
{[21000],[10-210],[-30201]}
Step 8
Step 8.1
List the vectors.
[21000],[10-210],[-30201]
Step 8.2
Write the vectors as a matrix.
[21-31000-22010001]
Step 8.3
To determine if the columns in the matrix are linearly dependent, determine if the equation Ax=0 has any non-trivial solutions.
Step 8.4
Write as an augmented matrix for Ax=0.
[21-3010000-22001000010]
Step 8.5
Find the reduced row echelon form.
Step 8.5.1
Multiply each element of R1 by 12 to make the entry at 1,1 a 1.
Step 8.5.1.1
Multiply each element of R1 by 12 to make the entry at 1,1 a 1.
[2212-320210000-22001000010]
Step 8.5.1.2
Simplify R1.
[112-32010000-22001000010]
[112-32010000-22001000010]
Step 8.5.2
Perform the row operation R2=R2-R1 to make the entry at 2,1 a 0.
Step 8.5.2.1
Perform the row operation R2=R2-R1 to make the entry at 2,1 a 0.
[112-3201-10-120+320-00-22001000010]
Step 8.5.2.2
Simplify R2.
[112-3200-123200-22001000010]
[112-3200-123200-22001000010]
Step 8.5.3
Multiply each element of R2 by -2 to make the entry at 2,2 a 1.
Step 8.5.3.1
Multiply each element of R2 by -2 to make the entry at 2,2 a 1.
[112-320-2⋅0-2(-12)-2(32)-2⋅00-22001000010]
Step 8.5.3.2
Simplify R2.
[112-32001-300-22001000010]
[112-32001-300-22001000010]
Step 8.5.4
Perform the row operation R3=R3+2R2 to make the entry at 3,2 a 0.
Step 8.5.4.1
Perform the row operation R3=R3+2R2 to make the entry at 3,2 a 0.
[112-32001-300+2⋅0-2+2⋅12+2⋅-30+2⋅001000010]
Step 8.5.4.2
Simplify R3.
[112-32001-3000-4001000010]
[112-32001-3000-4001000010]
Step 8.5.5
Perform the row operation R4=R4-R2 to make the entry at 4,2 a 0.
Step 8.5.5.1
Perform the row operation R4=R4-R2 to make the entry at 4,2 a 0.
[112-32001-3000-400-01-10+30-00010]
Step 8.5.5.2
Simplify R4.
[112-32001-3000-4000300010]
[112-32001-3000-4000300010]
Step 8.5.6
Multiply each element of R3 by -14 to make the entry at 3,3 a 1.
Step 8.5.6.1
Multiply each element of R3 by -14 to make the entry at 3,3 a 1.
[112-32001-30-14⋅0-14⋅0-14⋅-4-14⋅000300010]
Step 8.5.6.2
Simplify R3.
[112-32001-30001000300010]
[112-32001-30001000300010]
Step 8.5.7
Perform the row operation R4=R4-3R3 to make the entry at 4,3 a 0.
Step 8.5.7.1
Perform the row operation R4=R4-3R3 to make the entry at 4,3 a 0.
[112-32001-3000100-3⋅00-3⋅03-3⋅10-3⋅00010]
Step 8.5.7.2
Simplify R4.
[112-32001-30001000000010]
[112-32001-30001000000010]
Step 8.5.8
Perform the row operation R5=R5-R3 to make the entry at 5,3 a 0.
Step 8.5.8.1
Perform the row operation R5=R5-R3 to make the entry at 5,3 a 0.
[112-32001-30001000000-00-01-10-0]
Step 8.5.8.2
Simplify R5.
[112-32001-30001000000000]
[112-32001-30001000000000]
Step 8.5.9
Perform the row operation R2=R2+3R3 to make the entry at 2,3 a 0.
Step 8.5.9.1
Perform the row operation R2=R2+3R3 to make the entry at 2,3 a 0.
[112-3200+3⋅01+3⋅0-3+3⋅10+3⋅0001000000000]
Step 8.5.9.2
Simplify R2.
[112-3200100001000000000]
[112-3200100001000000000]
Step 8.5.10
Perform the row operation R1=R1+32R3 to make the entry at 1,3 a 0.
Step 8.5.10.1
Perform the row operation R1=R1+32R3 to make the entry at 1,3 a 0.
[1+32⋅012+32⋅0-32+32⋅10+32⋅00100001000000000]
Step 8.5.10.2
Simplify R1.
[112000100001000000000]
[112000100001000000000]
Step 8.5.11
Perform the row operation R1=R1-12R2 to make the entry at 1,2 a 0.
Step 8.5.11.1
Perform the row operation R1=R1-12R2 to make the entry at 1,2 a 0.
[1-12⋅012-12⋅10-12⋅00-12⋅00100001000000000]
Step 8.5.11.2
Simplify R1.
[10000100001000000000]
[10000100001000000000]
[10000100001000000000]
Step 8.6
Remove rows that are all zeros.
[100001000010]
Step 8.7
Write the matrix as a system of linear equations.
x=0
y=0
z=0
Step 8.8
Since the only solution to Ax=0 is the trivial solution, the vectors are linearly independent.
Linearly Independent
Linearly Independent
Step 9
Since the vectors are linearly independent, they form a basis for the null space of the matrix.
Basis of Nul(A): {[21000],[10-210],[-30201]}
Dimension of Nul(A): 3
