Find the Null Space [[1,-1,3],[5,-4,-2],[7,-6,4]]

Problem

Null Space*[[1,−1,3],[5,−4,−2],[7,−6,4]]

Solution

1. Set up the homogeneous equation A*x=0 to find the vectors x that satisfy the matrix equation.

[[1,−1,3],[5,−4,−2],[7,−6,4]]*[[(x_1)],[(x_2)],[(x_3)]]=[[0],[0],[0]]

2. Perform row operations to transform the matrix into row-echelon form. Subtract 5 times the first row from the second row ((R_2)−5*(R_1)⇒(R_2).

[[1,−1,3],[0,1,−17],[7,−6,4]]

3. Continue row reduction by subtracting 7 times the first row from the third row ((R_3)−7*(R_1)⇒(R_3).

[[1,−1,3],[0,1,−17],[0,1,−17]]

4. Eliminate the third row by subtracting the second row from the third row ((R_3)−(R_2)⇒(R_3).

[[1,−1,3],[0,1,−17],[0,0,0]]

5. Find the reduced row echelon form by adding the second row to the first row ((R_1)+(R_2)⇒(R_1).

[[1,0,−14],[0,1,−17],[0,0,0]]

6. Express the variables in terms of the free variable (x_3)

(x_1)−14*(x_3)=0⇒(x_1)=14*(x_3)

(x_2)−17*(x_3)=0⇒(x_2)=17*(x_3)

7. Write the solution in vector form by factoring out the free variable (x_3)

x=[[14*(x_3)],[17*(x_3)],[(x_3)]]=(x_3)*[[14],[17],[1]]

Final Answer

Null Space*([1,−1,3],[5,−4,−2],[7,−6,4])=span*{[14],[17],[1]}

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