Find the Null Space [[1,-1],[1,1]]

Problem

Null Space*[[1,−1],[1,1]]

Solution

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

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

2. Write the system of linear equations represented by the matrix multiplication.

(x_1)−(x_2)=0

(x_1)+(x_2)=0

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

[[1,−1,0],[1,1,0]]

(R_2)−(R_1)⇒(R_2)

[[1,−1,0],[0,2,0]]

4. Solve for the variables by continuing the reduction to reduced row echelon form.

1/2*(R_2)⇒(R_2)

[[1,−1,0],[0,1,0]]

(R_1)+(R_2)⇒(R_1)

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

5. Identify the solution for the variables (x_1) and (x_2)

(x_1)=0

(x_2)=0

6. Conclude the null space contains only the zero vector since there are no free variables.

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

Final Answer

Null Space*[[1,−1],[1,1]]={[0],[0]}

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