Find the Null Space [[-1/5,1/3],[2/5,-1/10]]

Problem

Null Space*[[−1/5,1/3],[2/5,−1/10]]

Solution

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

[[−1/5,1/3],[2/5,−1/10]]*[[(x_1)],[(x_2)]]=[[0],[0]]

2. Multiply the first row by −5 and the second row by 10 to clear the denominators and simplify the matrix.

[[1,−5/3],[4,−1]]

3. Perform row reduction by replacing the second row (R_2) with (R_2)−4*(R_1)

−1−4*(−5/3)=−1+20/3=17/3

[[1,−5/3],[0,17/3]]

4. Normalize the second row by multiplying it by 3/17

[[1,−5/3],[0,1]]

5. Eliminate the entry above the leading one in the second column by replacing (R_1) with (R_1)+5/3*(R_2)

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

6. Identify the solution to the system. Since the reduced row echelon form is the identity matrix, the only solution is the trivial solution.

(x_1)=0

(x_2)=0

Final Answer

Null Space*[[−1/5,1/3],[2/5,−1/10]]={[0],[0]}

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