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 ℝ^1

null(A)={0}

Final Answer

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

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