Find the Null Space A={[[1,2],[3,4]]}

Problem

A=[[1,2],[3,4]]

Solution

1. Set up the homogeneous equation A*x=0 where x=[[(x_1)],[(x_2)]] and 0=[[0],[0]]

[[1,2],[3,4]]*[[(x_1)],[(x_2)]]=[[0],[0]]

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

[[1,2,0],[3,4,0]]

3. Subtract 3 times the first row from the second row ((R_2)−3*(R_1)⇒(R_2).

[[1,2,0],[0,−2,0]]

4. Divide the second row by −2 (−1/2*(R_2)⇒(R_2).

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

5. Subtract 2 times the second row from the first row ((R_1)−2*(R_2)⇒(R_1) to reach reduced row-echelon form.

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

6. Interpret the resulting system of equations.

(x_1)=0

(x_2)=0

7. Conclude that since the only solution is the trivial solution, the null space contains only the zero vector.

Final Answer

Null(A)={[0],[0]}

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