Find the Null Space [[1,3],[7,-3]]

Problem

Null Space*[[1,3],[7,−3]]

Solution

1. Set up the equation for the null space by solving the homogeneous system A*x=0 where A is the given matrix.

[[1,3],[7,−3]]*[[(x_1)],[(x_2)]]=[[0],[0]]

2. Perform row reduction to find the row echelon form of the matrix. Subtract 7 times the first row from the second row.

(R_2)−7*(R_1)⇒(R_2)

[[1,3],[0,−24]]

3. Simplify the second row by dividing by −24 to obtain the reduced row echelon form.

−1/24*(R_2)⇒(R_2)

[[1,3],[0,1]]

4. Eliminate the entry above the pivot in the second column by subtracting 3 times the second row from the first row.

(R_1)−3*(R_2)⇒(R_1)

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

5. Identify the solution to the system. Since the reduced row echelon form is the identity matrix I the only solution to I*x=0 is the trivial solution.

(x_1)=0

(x_2)=0

6. Conclude the null space contains only the zero vector, meaning the nullity of the matrix is 0.

Null Space={[0],[0]}

Final Answer

Null Space*[[1,3],[7,−3]]={[0],[0]}

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