Find the Nullity [[-10,4],[5,-2],[20,-8],[-15,6]]

Problem

nullity*[[−10,4],[5,−2],[20,−8],[−15,6]]

Solution

1. Identify the matrix A and the definition of nullity. The nullity is the dimension of the null space of A which can be found using the Rank-Nullity Theorem: rank(A)+nullity(A)=n where n is the number of columns.

2. Determine the number of columns n The matrix has 2 columns, so n=2

3. Find the rank by performing row reduction to identify the number of linearly independent rows or columns. Observe that all rows are multiples of the second row [5,−2]

(R_1)=−2×[5,−2]

(R_3)=4×[5,−2]

(R_4)=−3×[5,−2]

4. Simplify the matrix to its row echelon form. Since all rows are proportional to a single non-zero row, there is only 1 pivot.

[[5,−2],[0,0],[0,0],[0,0]]

5. Calculate the rank. The number of non-zero rows in the row echelon form is 1, so rank(A)=1

6. Apply the formula for nullity.

nullity(A)=n−rank(A)

nullity(A)=2−1

nullity(A)=1

Final Answer

nullity*[[−10,4],[5,−2],[20,−8],[−15,6]]=1

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