Find the Null Space [[1,4,0,3,0],[0,0,1,-3,0],[0,0,0,0,-1]]

Problem

Null Space*[[1,4,0,3,0],[0,0,1,−3,0],[0,0,0,0,−1]]

Solution

1. Identify the matrix A and set up the homogeneous equation A*x=0 to find the null space.

A=[[1,4,0,3,0],[0,0,1,−3,0],[0,0,0,0,−1]]

2. Determine the pivot columns and free variables. The pivots are in columns 1, 3, and 5. The free variables correspond to columns 2 and 4, which we label (x_2) and (x_4)

(x_2)=s()

(x_4)=t

3. Write the system of equations corresponding to the row-reduced matrix.

(x_1)+4*(x_2)+3*(x_4)=0

(x_3)−3*(x_4)=0

−(x_5)=0

4. Solve for the pivot variables (x_1),(x_3),(x_5) in terms of the free variables s() and t

(x_1)=−4*s−3*t

(x_3)=3*t

(x_5)=0

5. Express the general solution vector x as a linear combination of vectors multiplied by the parameters s() and t

x=[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[−4*s−3*t],[s()],[3*t],[t],[0]]

x=s*[[−4],[1],[0],[0],[0]]+t*[[−3],[0],[3],[1],[0]]

Final Answer

Null Space*(A)=span*{[[−4],[1],[0],[0],[0]],[[−3],[0],[3],[1],[0]]}

Want more problems? Check here!