Find the Null Space [[1,2,3],[4,5,6],[7,8,9]]

Problem

Null Space*[[1,2,3],[4,5,6],[7,8,9]]

Solution

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

[[1,2,3],[4,5,6],[7,8,9]]*[[(x_1)],[(x_2)],[(x_3)]]=[[0],[0],[0]]

2. Perform row reduction to transform the matrix into row-echelon form. Subtract 4 times the first row from the second, and 7 times the first row from the third.

[[1,2,3],[0,−3,−6],[0,−6,−12]]

3. Simplify the rows by dividing the second row by −3

[[1,2,3],[0,1,2],[0,−6,−12]]

4. Eliminate the third row by adding 6 times the second row to the third row, and simplify the first row by subtracting 2 times the second row from it.

[[1,0,−1],[0,1,2],[0,0,0]]

5. Identify the free variable and write the system of equations. Here, (x_3) is a free variable t

(x_1)−(x_3)=0⇒(x_1)=t

(x_2)+2*(x_3)=0⇒(x_2)=−2*t

6. Express the solution in vector form by factoring out the parameter t

[[(x_1)],[(x_2)],[(x_3)]]=[[t],[−2*t],[t]]=t*[[1],[−2],[1]]

Final Answer

Null Space*[[1,2,3],[4,5,6],[7,8,9]]=span*{[1],[−2],[1]}

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