Find the Null Space [[1,0,4,0],[0,0,0,0],[0,0,0,0]]

Problem

Null Space*[[1,0,4,0],[0,0,0,0],[0,0,0,0]]

Solution

1. Identify the matrix A and the equation for the null space, which is the set of all vectors x such that A*x=0

A=[[1,0,4,0],[0,0,0,0],[0,0,0,0]]

2. Write the system of linear equations corresponding to A*x=0 using variables (x_1),(x_2),(x_3),(x_4)

1*(x_1)+0*(x_2)+4*(x_3)+0*(x_4)=0

3. Determine the pivot and free variables. The first column has a pivot in the first row, so (x_1) is a pivot variable. The variables (x_2),(x_3), and (x_4) are free variables.

(x_1)=−4*(x_3)

4. Express the general solution vector x in terms of the free variables (x_2),(x_3), and (x_4)

x=[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[−4*(x_3)],[(x_2)],[(x_3)],[(x_4)]]

5. Decompose the vector into a linear combination of basis vectors by factoring out each free variable.

x=(x_2)*[[0],[1],[0],[0]]+(x_3)*[[−4],[0],[1],[0]]+(x_4)*[[0],[0],[0],[1]]

6. State the basis for the null space as the set of these vectors.

Basis={[[0],[1],[0],[0]],[[−4],[0],[1],[0]],[[0],[0],[0],[1]]}

Final Answer

Null Space*[[1,0,4,0],[0,0,0,0],[0,0,0,0]]=span{[[0],[1],[0],[0]],[[−4],[0],[1],[0]],[[0],[0],[0],[1]]}

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