Find the Null Space

Problem

Null Space*[[0,4,−3,−1],[4,4,−2,−2],[−1,−1,1,−1],[0,2,1,3]]

Solution

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

A=[[0,4,−3,−1],[4,4,−2,−2],[−1,−1,1,−1],[0,2,1,3]]

2. Perform row operations to transform the matrix into reduced row echelon form (RREF). Swap the first and third rows and multiply by −1

[[1,1,−1,1],[4,4,−2,−2],[0,4,−3,−1],[0,2,1,3]]

3. Eliminate the first column by subtracting 4 times the first row from the second row.

[[1,1,−1,1],[0,0,2,−6],[0,4,−3,−1],[0,2,1,3]]

4. Swap the second and third rows to organize the pivots.

[[1,1,−1,1],[0,4,−3,−1],[0,0,2,−6],[0,2,1,3]]

5. Eliminate the second column by subtracting 0.5 times the second row from the fourth row.

[[1,1,−1,1],[0,4,−3,−1],[0,0,2,−6],[0,0,2.5,3.5]]

6. Eliminate the third column by subtracting 1.25 times the third row from the fourth row.

[[1,1,−1,1],[0,4,−3,−1],[0,0,2,−6],[0,0,0,11]]

7. Identify the rank of the matrix. Since there are 4 pivots (one in every row and column), the matrix is non-singular.

rank(A)=4

8. Apply the Rank-Nullity Theorem. Since the matrix is 4×4 and has rank 4, the dimension of the null space is 4 - 4 = 0$.

nullity(A)=0

Final Answer

Null Space*[[0,4,−3,−1],[4,4,−2,−2],[−1,−1,1,−1],[0,2,1,3]]={[0],[0],[0],[0]}

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