Find the Null Space [[0.4,0.1,0.5]]

Problem

[[0.4,0.1,0.5]]

Solution

1. Set up the equation for the null space by finding all vectors x such that A*x=0

x=[[(x_1)],[(x_2)],[(x_3)]]

0.4*(x_1)+0.1*(x_2)+0.5*(x_3)=0

2. Clear the decimals by multiplying the entire equation by 10 to simplify the coefficients.

4*(x_1)+(x_2)+5*(x_3)=0

3. Solve for one variable in terms of the others to identify the free variables.

(x_2)=−4*(x_1)−5*(x_3)

4. Express the solution vector in terms of the free variables (x_1) and (x_3)

[[(x_1)],[(x_2)],[(x_3)]]=[[(x_1)],[−4*(x_1)−5*(x_3)],[(x_3)]]

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

[[(x_1)],[(x_2)],[(x_3)]]=(x_1)*[[1],[−4],[0]]+(x_3)*[[0],[−5],[1]]

6. Identify the basis for the null space from the resulting vectors.

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

Final Answer

Null*([0.4,0.1,0.5])=span*{[[1],[−4],[0]],[[0],[−5],[1]]}

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