Find the Null Space {[[2],[1],[4]],[[-6],[0],[-1]]}

Problem

Null Space*{[[2],[1],[4]],[[−6],[0],[−1]]}

Solution

1. Construct the matrix A using the given vectors as rows to find the vectors x such that A*x=0

A=[[2,1,4],[−6,0,−1]]

2. Perform row operations to reach row-echelon form. Add 3 times the first row to the second row.

(R_2)+3*(R_1)⇒(R_2)

[[2,1,4],[0,3,11]]

3. Identify the pivot columns and the free variable. Columns 1 and 2 have pivots, so (x_3) is a free variable. Let (x_3)=t

4. Solve the system of equations starting from the second row.

3*(x_2)+11*t=0

(x_2)=−11/3*t

5. Substitute (x_2) and (x_3) into the first row equation to find (x_1)

2*(x_1)+(x_2)+4*(x_3)=0

2*(x_1)−11/3*t+4*t=0

2*(x_1)+1/3*t=0

(x_1)=−1/6*t

6. Express the solution in vector form by factoring out the parameter t

x=t*[[−1/6],[−11/3],[1]]

7. Simplify the basis vector by multiplying by 6 to remove fractions.

Basis=[[−1],[−22],[6]]

Final Answer

Null Space*{[[2],[1],[4]],[[−6],[0],[−1]]}=span*{[−1],[−22],[6]}

Want more problems? Check here!