Find the Nullity

Problem

Nullity*[[1,0,1,0,6],[1,1,2,2,0],[0,0,1,0,0],[0,0,0,0,0]]

Solution

1. Identify the matrix A and its dimensions.

A=[[1,0,1,0,6],[1,1,2,2,0],[0,0,1,0,0],[0,0,0,0,0]]

The matrix has m=4 rows and n=5 columns.

2. Perform row operations to find the Row Echelon Form (REF). Subtract row 1 from row 2 ((R_2)−(R_1)→(R_2).

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

3. Identify the pivots in the resulting matrix. The pivots are located in the first, second, and third columns.

Rank(A)=3

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

Rank(A)+Nullity(A)=n

5. Substitute the known values into the theorem (n=5 and Rank=3.

3+Nullity(A)=5

6. Solve for the nullity.

Nullity(A)=5−3

Nullity(A)=2

Final Answer

Nullity*[[1,0,1,0,6],[1,1,2,2,0],[0,0,1,0,0],[0,0,0,0,0]]=2

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