Find the Null Space [[20,16],[5,4],[-15,-12],[-20,-16]]

Problem

Null Space*[[20,16],[5,4],[−15,−12],[−20,−16]]

Solution

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

[[20,16],[5,4],[−15,−12],[−20,−16]]*[[(x_1)],[(x_2)]]=[[0],[0],[0],[0]]

2. Perform row reduction on the matrix to find its row echelon form. Divide the first row by 20

[[1,0.8],[5,4],[−15,−12],[−20,−16]]

3. Eliminate the entries below the pivot in the first column by performing (R_2)−5*(R_1) (R_3)+15*(R_1) and (R_4)+20*(R_1)

[[1,0.8],[0,0],[0,0],[0,0]]

4. Identify the free variable and write the equation for the non-zero row.

(x_1)+0.8*(x_2)=0

5. Express the pivot variable in terms of the free variable (x_2)

(x_1)=−0.8*(x_2)

6. Write the solution in vector form by factoring out the free variable (x_2)

x=[[−0.8*(x_2)],[(x_2)]]=(x_2)*[[−0.8],[1]]

7. Simplify the basis vector by multiplying by 5 to use integer components.

v=[[−4],[5]]

Final Answer

Null Space*[[20,16],[5,4],[−15,−12],[−20,−16]]=span*{[−4],[5]}

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