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 defining the matrix A and solving the homogeneous system A*x=0

A=[[1,1],[1,1]]

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

2. Perform row reduction to find the row reduced echelon form (RREF) of the matrix.

(R_2)−(R_1)→(R_2)

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

3. Identify the free variables and write the corresponding system of equations.

(x_1)+(x_2)=0

(x_1)=−(x_2)

4. Express the solution in vector form by setting the free variable (x_2)=t

x=[[−t],[t]]

x=t*[[−1],[1]]

5. Determine the basis for the null space, which is the set of vectors that span the solution set.

Null(A)=span*{[−1],[1]}

Final Answer

Null Space*[[1,1],[1,1]]={t*[[−1],[1]]:t∈ℝ}

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