Find the Null Space [[0,0],[-5,-8]]

Problem

Null Space*[[0,0],[−5,−8]]

Solution

1. Set up the equation for the null space by finding all vectors x=[[(x_1)],[(x_2)]] such that A*x=0

[[0,0],[−5,−8]]*[[(x_1)],[(x_2)]]=[[0],[0]]

2. Write the system of equations represented by the matrix multiplication.

0*(x_1)+0*(x_2)=0

−5*(x_1)−8*(x_2)=0

3. Solve for one variable in terms of the other using the non-trivial equation.

−5*(x_1)=8*(x_2)

(x_1)=−8/5*(x_2)

4. Express the solution vector in terms of the free variable (x_2)

[[(x_1)],[(x_2)]]=[[−8/5*(x_2)],[(x_2)]]

5. Factor out the free variable to find the basis vector for the null space.

[[(x_1)],[(x_2)]]=(x_2)*[[−8/5],[1]]

6. Simplify the basis vector by multiplying by a constant (5) to remove fractions, which does not change the span.

Basis=[[−8],[5]]

Final Answer

Null Space*[[0,0],[−5,−8]]=span*{[−8],[5]}

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