Find the Kernel

Problem

[[1,2,3,4],[0,0,1,2],[0,0,0,0]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0]]

Solution

1. Identify the system of linear equations from the matrix equation.

(x_1)+2*(x_2)+3*(x_3)+4*(x_4)=0

(x_3)+2*(x_4)=0

2. Determine the pivot and free variables. The pivots are in the first and third columns ((x_1) and (x_3), so (x_2) and (x_4) are free variables. Let (x_2)=s() and (x_4)=t

3. Solve for the pivot variables in terms of the free variables using back-substitution. From the second equation:

(x_3)=−2*(x_4)=−2*t

4. Substitute the expression for (x_3) into the first equation to solve for (x_1)

(x_1)+2*s+3*(−2*t)+4*t=0

(x_1)+2*s−6*t+4*t=0

(x_1)=−2*s+2*t

5. Express the general solution vector in terms of the parameters s() and t

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[−2*s+2*t],[s()],[−2*t],[t]]

6. Decompose the vector into a linear combination of basis vectors for the kernel.

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=s*[[−2],[1],[0],[0]]+t*[[2],[0],[−2],[1]]

Final Answer

ker*[[1,2,3,4],[0,0,1,2],[0,0,0,0]]=span{[[−2],[1],[0],[0]],[[2],[0],[−2],[1]]}

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