Find the Null Space [[1,7,0,6,0],[0,0,1,9,0],[0,0,0,0,8]]

Problem

null*[[1,7,0,6,0],[0,0,1,9,0],[0,0,0,0,8]]

Solution

1. Identify the pivot columns and free variables. The matrix is already in row echelon form. The pivots are in columns 1, 3, and 5. The free variables correspond to columns 2 and 4, which we denote as (x_2) and (x_4)

2. Set up the system of equations A*x=0 from the rows of the matrix.

(x_1)+7*(x_2)+6*(x_4)=0

(x_3)+9*(x_4)=0

8*(x_5)=0

3. Solve for the pivot variables (x_1),(x_3),(x_5) in terms of the free variables (x_2),(x_4)

(x_1)=−7*(x_2)−6*(x_4)

(x_3)=−9*(x_4)

(x_5)=0

4. Write the general solution vector x in parametric form by separating the free variables.

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[−7*(x_2)−6*(x_4)],[(x_2)],[−9*(x_4)],[(x_4)],[0]]

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=(x_2)*[[−7],[1],[0],[0],[0]]+(x_4)*[[−6],[0],[−9],[1],[0]]

5. Extract the basis vectors that span the null space.

Final Answer

null*[[1,7,0,6,0],[0,0,1,9,0],[0,0,0,0,8]]=span*{[[−7],[1],[0],[0],[0]],[[−6],[0],[−9],[1],[0]]}

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