Find the Null Space [[1,3,0,2],[0,0,1,9]]

Problem

Null Space*[[1,3,0,2],[0,0,1,9]]

Solution

1. Identify the matrix A and the equation for the null space, which is the set of all vectors x such that A*x=0

A=[[1,3,0,2],[0,0,1,9]]

2. Check the row echelon form. The matrix is already in reduced row echelon form (RREF) because the leading entries (pivots) are 1 and are the only non-zero entries in their respective columns.

[[1,3,0,2],[0,0,1,9]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0]]

3. Write the system of linear equations corresponding to the matrix rows.

(x_1)+3*(x_2)+2*(x_4)=0

(x_3)+9*(x_4)=0

4. Identify the free variables. Since the pivots are in columns 1 and 3, the free variables are (x_2) and (x_4) Let (x_2)=s() and (x_4)=t

(x_2)=s()

(x_4)=t

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

(x_1)=−3*s−2*t

(x_3)=−9*t

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

x=[[−3*s−2*t],[s()],[−9*t],[t]]

x=s*[[−3],[1],[0],[0]]+t*[[−2],[0],[−9],[1]]

7. Determine the basis for the null space, which consists of the vectors multiplied by the free variables.

Null(A)=span{[[−3],[1],[0],[0]],[[−2],[0],[−9],[1]]}

Final Answer

Null*[[1,3,0,2],[0,0,1,9]]=span{[[−3],[1],[0],[0]],[[−2],[0],[−9],[1]]}

Want more problems? Check here!