Find the Null Space [[2,1],[1,3]]

Problem

Null Space*[[2,1],[1,3]]

Solution

1. Set up the equation for the null space by defining the matrix A and solving the homogeneous system A*x=0

A=[[2,1],[1,3]]

[[2,1],[1,3]]*[[(x_1)],[(x_2)]]=[[0],[0]]

2. Perform row reduction on the augmented matrix to find the row echelon form.

[[2,1,0],[1,3,0]]

(R_1)↔(R_2)⇒[[1,3,0],[2,1,0]]

−2*(R_1)+(R_2)→(R_2)⇒[[1,3,0],[0,−5,0]]

3. Identify the pivot columns and determine if there are any free variables.

−1/5*(R_2)→(R_2)⇒[[1,3,0],[0,1,0]]

−3*(R_2)+(R_1)→(R_1)⇒[[1,0,0],[0,1,0]]

4. Solve for the variables based on the reduced row echelon form.

(x_1)=0

(x_2)=0

5. Conclude the null space based on the fact that only the trivial solution exists. Since the matrix is non-singular (its determinant is 6 - 1 = 5$), the null space contains only the zero vector.

Final Answer

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

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