Describe all solutions of Ax=0

Question

Let A= [1 -4  5  4 

            2 -8 10 8]

Describe all solutions of Ax=0. 

 

Answer 1

Let x = [x₁ x₂ x₃ x₄]^T with "T" meaning transpose.

 

Ax = 0 gives

 

x₁ - 4x₂ + 5x₃ + 4x₄ = 0 and

 

2x₁ - 8x₂ + 10x₃ + 8x₄ = 2(x₁ - 4x₂ + 5x₃ + 4x₄) = 0

 

 

There is only 1 independent equation describing x and it describes a plane in R⁴: x₁ - 4x₂ + 5x₃ + 4x₄ = 0.