Find the Kernel

Problem

ker(S)* where *S*([a],[b],[c])=([a+3*b−6*c],[2*a+b+c],[a+5*b+c])

Solution

1. Set up the system of linear equations by setting the output of the transformation equal to the zero vector.

a+3*b−6*c=0

2*a+b+c=0

a+5*b+c=0

2. Write the augmented matrix representing the system of equations.

([1,3,−6,0],[2,1,1,0],[1,5,1,0])

3. Perform row operations to reach row-echelon form. Subtract 2 times the first row from the second, and subtract the first row from the third.

([1,3,−6,0],[0,−5,13,0],[0,2,7,0])

4. Continue row reduction to find the relationship between variables. Multiply the second row by −1/5

([1,3,−6,0],[0,1,−2.6,0],[0,2,7,0])

5. Eliminate the second variable from the third row by subtracting 2 times the second row from the third.

([1,3,−6,0],[0,1,−2.6,0],[0,0,12.2,0])

6. Solve for the variables starting from the bottom row. The third row implies 12.2c = 0,s(o) = 0$.

c=0

7. Substitute back into the other equations. Since c=0 the second row b−2.6*c=0 implies b=0

b=0

8. Find the final variable using the first row a+3*b−6*c=0 Substituting b=0 and c=0 gives a=0

a=0

9. Identify the kernel as the set containing only the zero vector, since the only solution to the system is the trivial solution.

ker(S)={[0],[0],[0]}

Final Answer

ker(S)={[0],[0],[0]}

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