Find the Nullity

Problem

nullity*[[14,−18,−8,10,6,−6,−14],[−8,12,14,−4,−12,−10,10],[10,−14,−12,10,−12,4,16],[−6,10,16,−2,−14,−8,16],[12,−16,−10,8,8,18,6]]

Solution

1. Set up the matrix A and prepare to find its rank using Gaussian elimination.

A=[[14,−18,−8,10,6,−6,−14],[−8,12,14,−4,−12,−10,10],[10,−14,−12,10,−12,4,16],[−6,10,16,−2,−14,−8,16],[12,−16,−10,8,8,18,6]]

2. Reduce the matrix to row echelon form. Divide the first row by 2 and perform row operations to eliminate entries below the first pivot.

[[7,−9,−4,5,3,−3,−7],[0,12/7,66/7,12/7,−60/7,−94/7,14/7],[0,−8/7,−44/7,20/7,−114/7,58/7,182/7],[0,16/7,88/7,16/7,−80/7,−74/7,70/7],[0,−4/7,−22/7,−4/7,20/7,162/7,126/7]]

3. Simplify the rows. Notice that rows 2 and 4 are multiples of each other. Row 4 is 4/3 times row 2.

[[7,−9,−4,5,3,−3,−7],[0,1,11/2,1,−5,−47/6,7/6],[0,0,0,4,−22,−4/6,244/6],[0,0,0,0,0,7.33,4.33],[0,0,0,0,0,18.66,126.66]]

4. Identify the number of non-zero rows in the row echelon form. After full reduction, there are 5 linearly independent rows.

rank(A)=5

5. Apply the Rank-Nullity Theorem which states that for an m×n matrix, the rank plus the nullity equals the number of columns n

rank(A)+nullity(A)=n

6. Substitute the known values n=7 and rank(A)=5 into the formula.

nullity(A)=7−5

7. Calculate the final value.

nullity(A)=2

Final Answer

nullity*[[14,−18,−8,10,6,−6,−14],[−8,12,14,−4,−12,−10,10],[10,−14,−12,10,−12,4,16],[−6,10,16,−2,−14,−8,16],[12,−16,−10,8,8,18,6]]=2

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