Find the Kernel

Problem

[[1,−2,0,−10],[0,−1,1,−2],[0,0,2,4]]*x=[[0],[0],[0]]

Solution

1. Identify the task as finding the kernel (null space) of the given matrix A The kernel is the set of all vectors x such that A*x=0 Note that the vector [0,1,6]T provided in the prompt appears to be a specific output, but finding the kernel specifically requires solving the homogeneous system.

2. Set up the augmented matrix for the homogeneous system A*x=0

[[1,−2,0,−10,|,0],[0,−1,1,−2,|,0],[0,0,2,4,|,0]]

3. Simplify the third row by dividing by 2 to obtain a leading 1

[[1,−2,0,−10,|,0],[0,−1,1,−2,|,0],[0,0,1,2,|,0]]

4. Eliminate the entry above the leading 1 in the third column by performing (R_2)←(R_2)−(R_3)

[[1,−2,0,−10,|,0],[0,−1,0,−4,|,0],[0,0,1,2,|,0]]

5. Normalize the second row by multiplying by −1

[[1,−2,0,−10,|,0],[0,1,0,4,|,0],[0,0,1,2,|,0]]

6. Eliminate the entry above the leading 1 in the second column by performing (R_1)←(R_1)+2*(R_2)

[[1,0,0,−2,|,0],[0,1,0,4,|,0],[0,0,1,2,|,0]]

7. Express the variables (x_1),(x_2),(x_3) in terms of the free variable (x_4)

(x_1)−2*(x_4)=0⇒(x_1)=2*(x_4)

(x_2)+4*(x_4)=0⇒(x_2)=−4*(x_4)

(x_3)+2*(x_4)=0⇒(x_3)=−2*(x_4)

8. Write the solution in vector form by factoring out the free variable (x_4)

x=(x_4)*[[2],[−4],[−2],[1]]

Final Answer

ker*([1,−2,0,−10],[0,−1,1,−2],[0,0,2,4])=span*([2],[−4],[−2],[1])

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