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1. (10 points) In class we derived the optimal linear predictor for scalar data, and wrote down the optimal linear predictor for vector data (without proof). Here, let us derive the optimal linear predictor in the vector case. Suppose that {x1, x2, , xn) denote training data samples, where each Xi € Rd. Suppose {31, y2, , Yn } denote corresponding (scalar) labels. a. Show that the mean squared-error loss function for multivariate linear regression can be written in the following form: MSE(w) = ||y - Xww where X is an n X (d + 1) matrix and where the first column of X is all-ones. What is the dimension of W and y? What do the coordinates of W represent? b. Theoretically prove that the optimal linear regression weights are given by: w=(xix)-1X74. What algebraic assumptions on X did you make in order to derive the above closed form?

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