Find the Null Space

Problem

[[4,−1,0,−1,0,0,0,0,0,300],[−1,4,−1,0,−1,0,0,0,0,200],[0,−1,4,0,0,−1,0,0,0,300],[−1,0,0,4,−1,0,−1,0,0,100],[0,−1,0,−1,4,−1,0,−1,0,0],[0,0,−1,0,−1,4,0,0,−1,100],[0,0,0,−1,0,0,4,−1,0,160],[0,0,0,0,−1,0,−1,4,−1,60],[0,0,0,0,0,−1,0,−1,4,160]]

Solution

1. Set up the homogeneous system by defining the matrix A and solving the equation A*x=0 where x is a vector in ℝ10

2. Perform Gaussian elimination to transform the 9×10 matrix into its Reduced Row Echelon Form (RREF).

3. Identify the pivot columns and the free variable. Since there are 9 rows and 10 columns, and the rows are linearly independent, there is 1 free variable, which is (x_10)

4. Express the pivot variables in terms of the free variable (x_10) by reading the RREF rows as equations.

5. Write the solution in vector form by factoring out the free parameter (x_10) to find the basis for the null space.

6. Calculate the specific coefficients for the basis vector. Through row reduction, the relationship between the variables is determined to be:

(x_1)=−110*(x_10)

(x_2)=−100*(x_10)

(x_3)=−110*(x_10)

(x_4)=−80*(x_10)

(x_5)=−60*(x_10)

(x_6)=−80*(x_10)

(x_7)=−70*(x_10)

(x_8)=−60*(x_10)

(x_9)=−70*(x_10)

Final Answer

Null Space=span*{[−110],[−100],[−110],[−80],[−60],[−80],[−70],[−60],[−70],[1]}

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