Find the Null Space [[1,6,5,-7],[0,1,2,-1],[0,0,1,2]]

Problem

Null Space*[[1,6,5,−7],[0,1,2,−1],[0,0,1,2]]

Solution

1. Identify the matrix A and the system A*x=0 We are looking for the set of vectors x such that:

[[1,6,5,−7],[0,1,2,−1],[0,0,1,2]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0]]

2. Perform row reduction to find the Reduced Row Echelon Form (RREF). Start by eliminating the entry above the leading 1 in the third row. Subtract 2 times row 3 from row 2:

(R_2)−2*(R_3)→(R_2)

[[1,6,5,−7],[0,1,0,−5],[0,0,1,2]]

3. Continue row reduction by eliminating the entries above the leading 1 in the second row. Subtract 6 times row 2 from row 1:

(R_1)−6*(R_2)→(R_1)

[[1,0,5,23],[0,1,0,−5],[0,0,1,2]]

4. Finish row reduction by eliminating the entry above the leading 1 in the third row for the first row. Subtract 5 times row 3 from row 1:

(R_1)−5*(R_3)→(R_1)

[[1,0,0,13],[0,1,0,−5],[0,0,1,2]]

5. Write the equations corresponding to the RREF matrix:

(x_1)+13*(x_4)=0

(x_2)−5*(x_4)=0

(x_3)+2*(x_4)=0

6. Express the variables in terms of the free variable (x_4)

(x_1)=−13*(x_4)

(x_2)=5*(x_4)

(x_3)=−2*(x_4)

7. Write the solution vector in parametric form:

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[−13*(x_4)],[5*(x_4)],[−2*(x_4)],[(x_4)]]=(x_4)*[[−13],[5],[−2],[1]]

Final Answer

Null Space=span*{[−13],[5],[−2],[1]}

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