Find the Null Space [[1,2,0,0,2],[0,0,1,0,-1],[0,0,0,1,2]]

Problem

Null Space*[[1,2,0,0,2],[0,0,1,0,−1],[0,0,0,1,2]]

Solution

1. Identify the pivot columns and free variables. The matrix is already in reduced row echelon form (RREF). The pivots are in columns 1, 3, and 4. The free variables correspond to columns 2 and 5, which we label (x_2) and (x_5)

2. Write the system of equations corresponding to A*x=0

(x_1)+2*(x_2)+2*(x_5)=0

(x_3)−(x_5)=0

(x_4)+2*(x_5)=0

3. Express the pivot variables in terms of the free variables.

(x_1)=−2*(x_2)−2*(x_5)

(x_3)=(x_5)

(x_4)=−2*(x_5)

4. Write the general solution vector x by separating the components associated with each free variable.

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[−2*(x_2)−2*(x_5)],[(x_2)],[(x_5)],[−2*(x_5)],[(x_5)]]

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=(x_2)*[[−2],[1],[0],[0],[0]]+(x_5)*[[−2],[0],[1],[−2],[1]]

5. Define the null space as the span of these basis vectors.

Final Answer

Null Space=span*{[[−2],[1],[0],[0],[0]],[[−2],[0],[1],[−2],[1]]}

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