Find the Null Space [[-2,9,-6]]

Problem

Null Space*[[−2,9,−6]]

Solution

1. Identify the matrix A and the definition of the null space. The null space consists of all vectors x such that A*x=0

A=[[−2,9,−6]]

2. Set up the homogeneous equation A*x=0 for a vector x in ℝ3

[[−2,9,−6]]*[[(x_1)],[(x_2)],[(x_3)]]=0

3. Write the equation in algebraic form.

−2*(x_1)+9*(x_2)−6*(x_3)=0

4. Solve for the pivot variable (x_1) in terms of the free variables (x_2) and (x_3)

−2*(x_1)=−9*(x_2)+6*(x_3)

(x_1)=9/2*(x_2)−3*(x_3)

5. Express the general solution vector x in terms of the free variables.

[[(x_1)],[(x_2)],[(x_3)]]=[[9/2*(x_2)−3*(x_3)],[(x_2)],[(x_3)]]

6. Decompose the vector into a linear combination of basis vectors by factoring out the free variables.

[[(x_1)],[(x_2)],[(x_3)]]=(x_2)*[[9/2],[1],[0]]+(x_3)*[[−3],[0],[1]]

7. Identify the basis for the null space from the resulting vectors. To avoid fractions, the first vector can be multiplied by 2.

Basis={[[9],[2],[0]],[[−3],[0],[1]]}

Final Answer

Null Space*[[−2,9,−6]]=span{[[9],[2],[0]],[[−3],[0],[1]]}

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