Yeah. Okay. Um so suppose were given a basis of two vectors, which is uh first factors 1 -1. The second vector is 1, 1. And uh were also given a W, which is another factor that is equal to 10 And I want to find the coordinate factor of W. With respect to this basis. Okay, How we do that? Well, let us reformulate this to an easier to handle form. So I'm going to collate the basis factors into this one matrix over here, and I'm going to multiply it by some arbitrary uh you know, like uh column vector. Let's call that X. Y. And then that's going to end up using us back the W which is 10 Okay, so finding the coordinate vector of W. With respect to this basis. Okay. Is the same as finding this column vector X. Y. Under this matrix multiplication. This this problem that I've set up over here. Okay, so how do we solve for X and Y? Well, We just rearrange the terms. Okay, that's equal to 1. -1. 1. The inverse of that Gay Times. 1, 0. Okay. How do we find the inverse of a two by two matrix? Well, again, let us recall that if I have a two by two matrix, A, B C. D. And I want to find its inverse. Well, that's going to be equal to one over the determinant of this matrix, which is a d minus Bc. Okay, and then for the actual matrix I need to exchange the diagonals so it's D. A. And then the off diagonal stays where they are. But I have to put a negative sign in front so negative b negative. See so this is the formula for finding the inverse of a two by two matrix. Okay, so over here I'm going to apply this formula. The determinant of this is one minus minus one. So it's just too, so it's one over to over here and then I have to exchange the diagnose which is just 11 Uh And then over here I have to put a negative side in front of the off dying. And also I have negative 11 Okay. And then I multiplied by the same column vector 10 Well, what is this going to give us, this is going to give us the vector 1/2, 1/2. And therefore we are concluding that the coordinate vector of W with respect to this basis is going to be one half, one half. Okay, so let's clear the screen and work on the second example. So I suppose this time the first basis vector is one minus one, and the second basis factor is 11 And the w. Okay, that we're given is 01 Okay well how would we go about finding the coordinate vector of W. With respect to this basis. While we do the same thing, we're going to collate the basis factors into this one big matrix, multiply it to the column vector X. Y. And set it equal to W. And to finally quote in that vector of W. With respect to the spaces. We just have to solve for X. Y. Well how do we do this? Well we have X. Y. Is equal to 1 -1, 1, 1, Inverse Times 01. And how do we do this? Well again we take the determinant, so 1 -1. So it's 1/2. Okay. And then what is that? Um how do we figure out the matrix part? Well the diagnose have to be exchanged but then they're both ones so they just stay, you know they just stay the same as 11 and they have to put negative signs in front of the off diagnose. So I have minus 11 like this And then we're going to have a zero one. Okay? So uh if you were to multiply this thing out, uh this picks out the second column of this matrix, so this just becomes minus one half in one half. Okay. So therefore we are concluding that minus one half one have this column vector is going to be the coordinate factor of W with respect to this basis uh that I've shown here on the left.