Find the Null Space [[1,2],[4,8]]

Problem

Null Space*[[1,2],[4,8]]

Solution

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

[[1,2],[4,8]]*[[(x_1)],[(x_2)]]=[[0],[0]]

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

(R_2)−4*(R_1)⇒(R_2)

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

3. Identify the free variables by looking for columns without pivots.

(x_1)+2*(x_2)=0

(x_2)=t

4. Solve for the pivot variables in terms of the free variables.

(x_1)=−2*t

5. Express the solution in vector form to define the basis of the null space.

[[(x_1)],[(x_2)]]=[[−2*t],[t]]=t*[[−2],[1]]

Final Answer

Null Space*[[1,2],[4,8]]=span*{[−2],[1]}

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