Find the Null Space [[7,6,2],[6,3,1],[5,5,6]]

Problem

Null Space*[[7,6,2],[6,3,1],[5,5,6]]

Solution

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

A=[[7,6,2],[6,3,1],[5,5,6]]

2. Perform row operations to transform the matrix into row-echelon form. Subtract the second row from the first row.

(R_1)−(R_2)→(R_1)

[[1,3,1],[6,3,1],[5,5,6]]

3. Eliminate entries in the first column below the pivot.

−6*(R_1)+(R_2)→(R_2)

−5*(R_1)+(R_3)→(R_3)

[[1,3,1],[0,−15,−5],[0,−10,1]]

4. Simplify the second row by dividing by −5

−1/5*(R_2)→(R_2)

[[1,3,1],[0,3,1],[0,−10,1]]

5. Eliminate the entry in the third row, second column.

10/3*(R_2)+(R_3)→(R_3)

[[1,3,1],[0,3,1],[0,0,13/3]]

6. Identify the rank of the matrix. Since there are three pivots, the rank is 3.

rank(A)=3

7. Apply the Rank-Nullity Theorem to find the dimension of the null space.

nullity(A)=n−rank(A)

nullity(A)=3−3=0

8. Conclude that since the matrix is non-singular (the determinant is non-zero), the only solution to A*x=0 is the zero vector.

Final Answer

Null Space*[[7,6,2],[6,3,1],[5,5,6]]={[0],[0],[0]}

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