Find the Null Space

Problem

A=[[1,−4,0,4,0],[0,0,1,5,0],[0,0,0,0,−1]]

Solution

1. Identify the pivot columns and free variables by looking at the matrix 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 from the matrix equation A*x=0

(x_1)−4*(x_2)+4*(x_4)=0

(x_3)+5*(x_4)=0

−(x_5)=0

3. Express the basic variables in terms of the free variables (x_2) and (x_4)

(x_1)=4*(x_2)−4*(x_4)

(x_3)=−5*(x_4)

(x_5)=0

4. Write the general solution vector x as a linear combination of vectors scaled by the free variables.

x=[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[4*(x_2)−4*(x_4)],[(x_2)],[−5*(x_4)],[(x_4)],[0]]

x=(x_2)*[[4],[1],[0],[0],[0]]+(x_4)*[[−4],[0],[−5],[1],[0]]

5. Identify the basis for the null space from the vectors multiplied by the free variables.

Final Answer

Null(A)=span*{[[4],[1],[0],[0],[0]],[[−4],[0],[−5],[1],[0]]}

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