 As I said, we're going to expand on this a bit. I just want to show you if we have a number e to the power This complex number alpha plus i beta or beta i t. Remember that would be e to the power alpha t plus i beta t So that's that which is the same as e to the power alpha t times e to the power i times beta t And I can rewrite that that can we can remember we can always read like this e to the power alpha t and then we have the cosine of beta t plus i times the sine of Beta t we can always write that as such Now remember in our previous example. We had this eigenvector We had one and one of a negative one remember I can I can rewrite this remember this is one plus zero I there so I can rewrite this as Let's have and I can't remember that was exactly the What we had there, but I can write it as one and one plus zero and a negative two I can have the i there or the i in front That will be exactly the same. It's one plus nothing. It's just a one and one negative two i That's exactly the same there and this for this complex eigenvector. We call b1 So b1 is the real part of k1 and This we call b2 and that is the imaginary part of k1 And remember because I said that Case of two equals the complex conjugate of case of one and lambda sub two equals the complex conjugate of lambda sub one I'm always going to have this same Form of the general solution set and what we're going to have it is this is always going to be an x1 And x2 for two by two two by two matrix It's e to the power alpha t then betas abyss of one Remember that's the real part of k1 and b sub 2 is the imaginary part of k1 So it's b sub 1 cosine of beta t minus b sub 2 sine of beta t and x sub 2 is going to be times e to the power alpha t now beta 2 cosine beta t plus there's a negative there's a positive there's b1 There's b2 positive beta sub one abyss of one sine of beta t don't get the b's and betas Mixed up so you always going to have that and then your final set of solutions These are not linear these are not linear The payments is going to be c times x sub 1 plus c sub 2 x sub 2 c times c Sub 1 x sub 1 plus c sub 2 x sub 2