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

Problem

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

Solution

1. Set up the homogeneous system A*x=0 by representing the matrix as an augmented matrix.

[[1,−1,3,0],[5,−4,−4,0],[7,−6,2,0]]

2. Perform row operations to reach row echelon form. Subtract 5 times the first row from the second row, and 7 times the first row from the third row.

[[1,−1,3,0],[0,1,−19,0],[0,1,−19,0]]

3. Simplify further by subtracting the second row from the third row to create a row of zeros.

[[1,−1,3,0],[0,1,−19,0],[0,0,0,0]]

4. Find the reduced row echelon form by adding the second row to the first row.

[[1,0,−16,0],[0,1,−19,0],[0,0,0,0]]

5. Identify the free variable and write the solution in parametric form. Here, (x_3) is free. Let (x_3)=t

(x_1)−16*t=0⇒(x_1)=16*t

(x_2)−19*t=0⇒(x_2)=19*t

6. Express the solution as a vector space spanned by the basis vector.

[[(x_1)],[(x_2)],[(x_3)]]=t*[[16],[19],[1]]

Final Answer

Null Space*[[1,−1,3],[5,−4,−4],[7,−6,2]]=span*{[16],[19],[1]}

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