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

Problem

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

Solution

1. Set up the equation for the null space by looking for all vectors x=[[(x_1)],[(x_2)]] such that A*x=0

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

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

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

3. Eliminate the entry in the second row, first column by replacing (R_2) with (R_2)−2*(R_1)

[[1,3,0],[0,−5,0]]

4. Scale the second row by dividing by −5 to obtain the reduced row echelon form.

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

5. Back-substitute to solve for the variables. The second row gives (x_2)=0 Substituting this into the first row (x_1)+3*(x_2)=0 gives (x_1)=0

(x_1)=0

(x_2)=0

6. Identify the basis of the null space. Since the only solution is the zero vector, the null space contains only the origin.

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

Final Answer

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

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