Find the Null Space [[1,0,-4,9],[0,1,1,-7],[0,0,0,0]]

Problem

Null Space*[[1,0,−4,9],[0,1,1,−7],[0,0,0,0]]

Solution

1. Identify the pivot columns and free variables in the matrix. The matrix is already in reduced row echelon form (RREF). The pivots are in the first and second columns, corresponding to variables (x_1) and (x_2) The free variables are (x_3) and (x_4)

2. Write the system of equations corresponding to the non-zero rows of the matrix.

(x_1)−4*(x_3)+9*(x_4)=0

(x_2)+(x_3)−7*(x_4)=0

3. Express the pivot variables in terms of the free variables (x_3) and (x_4)

(x_1)=4*(x_3)−9*(x_4)

(x_2)=−(x_3)+7*(x_4)

4. Write the general solution vector x in terms of the free variables.

x=[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[4*(x_3)−9*(x_4)],[−(x_3)+7*(x_4)],[(x_3)],[(x_4)]]

5. Decompose the vector into a linear combination of vectors scaled by the free variables to find the basis for the null space.

x=(x_3)*[[4],[−1],[1],[0]]+(x_4)*[[−9],[7],[0],[1]]

Final Answer

Null Space=span*{[[4],[−1],[1],[0]],[[−9],[7],[0],[1]]}

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