Find the Null Space [[2],[1],[4]]

Problem

null*[[2],[1],[4]]

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],[1],[4]]

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

[[2],[1],[4]]*[(x_1)]=[[0],[0],[0]]

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

2*(x_1)=0

1*(x_1)=0

4*(x_1)=0

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

(x_1)=0

5. Conclude that the only vector in the null space is the zero vector. Since there are no free variables, the null space contains only the origin of the domain ℝ1

null(A)={0}

Final Answer

null*[[2],[1],[4]]={0}

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