Find the Null Space [[5,-8],[-10,16]]

Problem

Null Space*[[5,−8],[−10,16]]

Solution

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

[[5,−8],[−10,16]]*[[(x_1)],[(x_2)]]=[[0],[0]]

2. Reduce the matrix to row-echelon form by performing the row operation (R_2)+2*(R_1)→(R_2)

[[5,−8],[0,0]]

3. Identify the pivot and free variables. The first column has a pivot, so (x_1) is a basic variable. The second column has no pivot, so (x_2) is a free variable.

(x_2)=t

4. Solve for the basic variable (x_1) in terms of the free variable t using the first row of the reduced matrix.

5*(x_1)−8*t=0

5*(x_1)=8*t

(x_1)=8/5*t

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

x=[[8/5*t],[t]]=t*[[8/5],[1]]

6. Simplify the basis vector by choosing t=5 to eliminate the fraction.

Basis=[[8],[5]]

Final Answer

Null Space*([5,−8],[−10,16])=span*{[8],[5]}

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