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[MUSIC] So weve been solving this differential equation Ẋ = Ax. A is a two-by-two matrix. X is a column vector X1 and X2. In the next series of lectures, I want to show you how to visualize the solution of this equation. Those diagrams are called phase portraits and the visualization is done in whats called the phase space of the solution. So X is a column vector X1 and X2 and the visualization then draws the solution of X2 versus X1. So first we have to say something about this solution, the origin here. The 00 value is called a fixed point. So thats the point right at the origin. The meaning of a fixed point is that if the solution starts at 00. So if the initial condition for X was 00 then it stays at 00 for all time. So the solution is fixed at 00 because the derivative is 0. So the origin is considered a fixed point. Its also sometimes called an equilibrium point of the equation and the equilibrium point can be stable or unstable. So if an equilibri