Find the Null Space

Problem

S={[[2],[−1],[3],[4],[2]],[[6],[2],[−1],[3],[4]]}

Solution

1. Construct the matrix A using the given vectors as rows, since the null space is the set of all vectors x such that A*x=0

A=[[2,−1,3,4,2],[6,2,−1,3,4]]

2. Perform row reduction to find the row echelon form. Subtract 3 times the first row from the second row.

(R_2)→(R_2)−3*(R_1)

[[2,−1,3,4,2],[0,5,−10,−9,−2]]

3. Normalize the pivot rows. Divide the first row by 2 and the second row by 5.

[[1,−0.5,1.5,2,1],[0,1,−2,−1.8,−0.4]]

4. Eliminate the term above the second pivot. Add 0.5 times the second row to the first row to reach reduced row echelon form (RREF).

(R_1)→(R_1)+0.5*(R_2)

[[1,0,0.5,1.1,0.8],[0,1,−2,−1.8,−0.4]]

5. Identify the free variables. Since there are pivots in columns 1 and 2, the free variables are (x_3) (x_4) and (x_5)

(x_1)+0.5*(x_3)+1.1*(x_4)+0.8*(x_5)=0

(x_2)−2*(x_3)−1.8*(x_4)−0.4*(x_5)=0

6. Express the pivot variables in terms of the free variables.

(x_1)=−0.5*(x_3)−1.1*(x_4)−0.8*(x_5)

(x_2)=2*(x_3)+1.8*(x_4)+0.4*(x_5)

7. Write the general solution in vector form by setting each free variable to 1 and the others to 0.

(v_1)=[[−0.5],[2],[1],[0],[0]],(v_2)=[[−1.1],[1.8],[0],[1],[0]],(v_3)=[[−0.8],[0.4],[0],[0],[1]]

Final Answer

Null(S)=span*{[[−0.5],[2],[1],[0],[0]],[[−1.1],[1.8],[0],[1],[0]],[[−0.8],[0.4],[0],[0],[1]]}

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