Find the Null Space [[1,2,-3,-1],[-2,-4,6,3]]

Problem

Null Space*[[1,2,−3,−1],[−2,−4,6,3]]

Solution

1. Set up the homogeneous system by representing the matrix as A and solving the equation A*x=0

A=[[1,2,−3,−1],[−2,−4,6,3]]

2. Perform row reduction to find the Row Reduced Echelon Form (RREF) of the matrix.

(R_2)+2*(R_1)⇒(R_2)

[[1,2,−3,−1],[0,0,0,1]]

3. Identify the pivot and free variables based on the leading ones in the RREF.

Pivots: *(x_1),(x_4)

Free variables: *(x_2),(x_3)

4. Solve for the pivot variables in terms of the free variables using the reduced equations.

(x_4)=0

(x_1)+2*(x_2)−3*(x_3)−(x_4)=0⇒(x_1)=−2*(x_2)+3*(x_3)

5. Express the solution vector x in parametric form by separating the free variables.

x=[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[−2*(x_2)+3*(x_3)],[(x_2)],[(x_3)],[0]]

x=(x_2)*[[−2],[1],[0],[0]]+(x_3)*[[3],[0],[1],[0]]

6. State the basis for the null space, which consists of the vectors multiplied by the free variables.

Basis={[[−2],[1],[0],[0]],[[3],[0],[1],[0]]}

Final Answer

Null Space*[[1,2,−3,−1],[−2,−4,6,3]]=span{[[−2],[1],[0],[0]],[[3],[0],[1],[0]]}

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