Find the Nullity

Problem

nullity*[[1,1,−2,0,1,−3],[1,2,−3,0,−3,−2],[1,3,−4,0,−4,5],[1,3,−3,1,0,−3],[1,4,−5,0,−8,14]]

Solution

1. Set up the matrix A representing the given system.

A=[[1,1,−2,0,1,−3],[1,2,−3,0,−3,−2],[1,3,−4,0,−4,5],[1,3,−3,1,0,−3],[1,4,−5,0,−8,14]]

2. Perform row operations to reach row echelon form. Subtract the first row from all other rows.

[[1,1,−2,0,1,−3],[0,1,−1,0,−4,1],[0,2,−2,0,−5,8],[0,2,−1,1,−1,0],[0,3,−3,0,−9,17]]

3. Continue row reduction by using the second row to eliminate entries in the second column.

[[1,1,−2,0,1,−3],[0,1,−1,0,−4,1],[0,0,0,0,3,6],[0,0,1,1,7,−2],[0,0,0,0,3,14]]

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

[[1,1,−2,0,1,−3],[0,1,−1,0,−4,1],[0,0,1,1,7,−2],[0,0,0,0,3,6],[0,0,0,0,3,14]]

5. Eliminate the entry in the fifth row using the fourth row.

[[1,1,−2,0,1,−3],[0,1,−1,0,−4,1],[0,0,1,1,7,−2],[0,0,0,0,3,6],[0,0,0,0,0,8]]

6. Identify the rank by counting the number of non-zero rows (pivots). There are 5 pivots.

rank(A)=5

7. Apply the Rank-Nullity Theorem which states that rank(A)+nullity(A)=n where n is the number of columns.

5+nullity(A)=6

8. Solve for the nullity.

nullity(A)=6−5=1

Final Answer

nullity*[[1,1,−2,0,1,−3],[1,2,−3,0,−3,−2],[1,3,−4,0,−4,5],[1,3,−3,1,0,−3],[1,4,−5,0,−8,14]]=1

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