Find the Null Space [[12],[3],[-9]]

Problem

Null Space*[[12],[3],[−9]]

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=[[12],[3],[−9]]

2. Set up the homogeneous equation A*x=0 Since A is a 3×1 matrix, x must be a 1×1 vector (a scalar) (x_1)

[[12],[3],[−9]]*[(x_1)]=[[0],[0],[0]]

3. Write the resulting system of linear equations from the matrix multiplication.

12*(x_1)=0

3*(x_1)=0

−9*(x_1)=0

4. Solve for the variable (x_1) In all three equations, the only solution is (x_1)=0

(x_1)=0

5. Conclude that the only vector in the null space is the zero vector.

Null(A)={0}

Final Answer

Null Space*[[12],[3],[−9]]={[[*0*]]}

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