Find the Null Space s=[[2,4,6,4],[2,5,7,6],[2,3,5,2]]

Problem

null*[[2,4,6,4],[2,5,7,6],[2,3,5,2]]

Solution

1. Set up the augmented matrix for the homogeneous system A*x=0 to find the vectors x that satisfy the equation.

[[2,4,6,4,0],[2,5,7,6,0],[2,3,5,2,0]]

2. Perform row operations to reach row echelon form. Subtract the first row from the second and third rows.

(R_2)−(R_1)⇒(R_2)

(R_3)−(R_1)⇒(R_3)

[[2,4,6,4,0],[0,1,1,2,0],[0,−1,−1,−2,0]]

3. Continue row reduction to reach reduced row echelon form (RREF). Add the second row to the third row, and divide the first row by 2.

(R_3)+(R_2)⇒(R_3)

1/2*(R_1)⇒(R_1)

[[1,2,3,2,0],[0,1,1,2,0],[0,0,0,0,0]]

4. Eliminate the pivot column above the second row pivot by subtracting twice the second row from the first row.

(R_1)−2*(R_2)⇒(R_1)

[[1,0,1,−2,0],[0,1,1,2,0],[0,0,0,0,0]]

5. Identify free variables and write the general solution. Here, (x_3) and (x_4) are free variables.

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

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

6. Express the solution in vector form by separating the components associated with each free variable.

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=(x_3)*[[−1],[−1],[1],[0]]+(x_4)*[[2],[−2],[0],[1]]

Final Answer

null*[[2,4,6,4],[2,5,7,6],[2,3,5,2]]=span*{[[−1],[−1],[1],[0]],[[2],[−2],[0],[1]]}

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