Find the Null Space [[0,10,20,30],[25,45,65,85]]

Problem

Null Space*[[0,10,20,30],[25,45,65,85]]

Solution

1. Set up the homogeneous system by representing the matrix as A and solving the equation A*x=0 for the vector x=[(x_1),(x_2),(x_3),(x_4)]T

2. Simplify the rows by dividing the first row by 10 and the second row by 5 to make the numbers more manageable.

[[0,1,2,3],[5,9,13,17]]

3. Perform row operations to reach row-echelon form. Swap the first and second rows.

[[5,9,13,17],[0,1,2,3]]

4. Reduce to reduced row-echelon form (RREF) by performing (R_1)←(R_1)−9*(R_2)

[[5,0,−5,−10],[0,1,2,3]]

5. Normalize the first row by dividing (R_1) by 5

[[1,0,−1,−2],[0,1,2,3]]

6. Identify the pivot and free variables. The pivots are in columns 1 and 2 ((x_1),(x_2), and the free variables are (x_3) and (x_4)

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

(x_1)=(x_3)+2*(x_4)

(x_2)=−2*(x_3)−3*(x_4)

8. Write the general solution in vector form by separating the components associated with (x_3) and (x_4)

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

Final Answer

Null Space*[[0,10,20,30],[25,45,65,85]]=span{[[1],[−2],[1],[0]],[[2],[−3],[0],[1]]}

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