We're just going to actually execute the commands here in a particular data set for predicting an outcome. Using the linear combinations of a couple of vectors from a design matrix. So I'm gonna use the data set mtcars, I'm gonna do data(mtcars). And then here's the first couple of rows and I want my y to be mtcars miles per gallon. And I want my x to be a matrix that has a vector of 1s, a weight vector And horse power. Okay, so let's just check head(x). Okay, now we know that the best predictor of y, that's created as the collection on linear combinations of the columns of x, is solve Inverse of x transpose x, times X transpose times y. And there it is. We can check this with our lm function. We have miles per gallon as the outcome and we had weight as a predictor and horse power as a predictor. And this is what the lm function does automatically, let me just grab the coefficients. And you can see, they're identical, okay? So that is our equation, let's write it out one more time. Our equation, if we have a vector y, an n by 1 vector y that we're interested in predicting with an n by p matrix x. And we wanna predict y or explain y with the linear combinations of the columns of x. The best estimator of the collection of column weights is x transpose x, inverse x transpose y. And we just saw that when we did that, we got these collection of numbers throughout the class. Now we're gonna talk a lot more about how to use those numbers and more about their statistical properties.