Find the Null Space

Problem

[[4,3,−3,−7],[−8,−3,15,−1],[16,3,−39,17]]

Solution

1. Set up the equation for the null space by solving the homogeneous system A*x=0 where x=[(x_1),(x_2),(x_3),(x_4)]T

2. Perform row operations to transform the matrix into row-echelon form. Add 2 times the first row to the second row ((R_2)+2*(R_1)→(R_2) and subtract 4 times the first row from the third row ((R_3)−4*(R_1)→(R_3).

[[4,3,−3,−7],[0,3,9,−15],[0,−9,−27,45]]

3. Continue row reduction by adding 3 times the second row to the third row ((R_3)+3*(R_2)→(R_3).

[[4,3,−3,−7],[0,3,9,−15],[0,0,0,0]]

4. Simplify to reduced row-echelon form by dividing the second row by 3 ((R_2)/3→(R_2) and then subtracting the new second row from the first row ((R_1)−(R_2)→(R_1). Finally, divide the first row by 4 ((R_1)/4→(R_1).

[[1,0,−3,2],[0,1,3,−5],[0,0,0,0]]

5. Identify free variables and express the pivot variables (x_1) and (x_2) in terms of the free variables (x_3) and (x_4)

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

(x_2)+3*(x_3)−5*(x_4)=0⇒(x_2)=−3*(x_3)+5*(x_4)

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

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=(x_3)*[[3],[−3],[1],[0]]+(x_4)*[[−2],[5],[0],[1]]

Final Answer

Null Space=span*{[[3],[−3],[1],[0]],[[−2],[5],[0],[1]]}

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