Find the Null Space [[8,9]]

Problem

Null Space*[[8,9]]

Solution

1. Identify the matrix A=[[8,9]] which represents a linear transformation from ℝ2 to ℝ1

2. Set up the homogeneous equation A*x=0 for a vector x=[[(x_1)],[(x_2)]]

8*(x_1)+9*(x_2)=0

3. Solve for one variable in terms of the other to find the general solution.

8*(x_1)=−9*(x_2)

(x_1)=−9/8*(x_2)

4. Express the solution in vector form by treating (x_2) as a free variable t

[[(x_1)],[(x_2)]]=[[−9/8*t],[t]]

5. Factor out the scalar t to find the basis vector for the null space.

[[(x_1)],[(x_2)]]=t*[[−9/8],[1]]

6. Simplify the basis vector by multiplying by 8 to eliminate fractions, which does not change the span.

Basis=[[−9],[8]]

Final Answer

Null Space*[[8,9]]=span*{[−9],[8]}

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