Find the Null Space s={[[-7],[5],[1]],[[-6],[5],[0]]}

Problem

S={[[−7],[5],[1]],[[−6],[5],[0]]}

Solution

1. Identify the task as finding the null space of a matrix A where the given vectors are the columns of that matrix.

2. Construct the matrix A using the given vectors as columns.

A=[[−7,−6],[5,5],[1,0]]

3. Set up the homogeneous equation A*x=0 to find the null space, where x=[[(x_1)],[(x_2)]]

[[−7,−6],[5,5],[1,0]]*[[(x_1)],[(x_2)]]=[[0],[0],[0]]

4. Solve the system of linear equations. From the third row, we have 1*(x_1)+0*(x_2)=0

(x_1)=0

5. Substitute (x_1)=0 into the second row equation 5*(x_1)+5*(x_2)=0

5*(0)+5*(x_2)=0

(x_2)=0

6. Conclude that the only solution to the system is the trivial solution x=[[0],[0]] Since the columns are linearly independent, the null space contains only the zero vector.

Final Answer

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

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