IFT6135-H2019 - Assignment 0 - Linear Algebra and Probability

IFT6135-H2019

Prof : Aaron Courville[pic 1]

Due Date : January 25 (10pm), 2019

Instructions

Assignment 0 Linear Algebra and Probability

This assignment serves as a warm-up for the following assignments. You are not obliged to finish this assignment, but some of the results here might be useful for the upcoming assign- ments. Unless otherwise specified, you may use the results in this assignment directly in your answer in the future.[pic 2]

• Use a document preparation system such as LaTeX.

You will be using Gradescope, you should have received an email to sign up, otherwise sign up for an account on gradescope.com and use course code 9EVRGV[pic 3]

• Submit this test submission on Gradescope (not necessary to complete and not marked)

Question 1. Given any unit vector n (i.e. ||n|| = 1), we define the hyperplane Hn := {x : nTx = 0} for which n is known as the normal vector. For any vector x, we define its projection into Hn as πn(x) = x − (xTn)n.

1. Given two vectors x = x , take n = x2−x1 . Show that π (x ) = π (x ).[pic 4]

||x2−x1||

1. Let   w  be a vector and define y1 := xT1 w  and y2 := xT2 w. Show that y1 = y2 if and only if

w ∈ Hn.

∗3.Let X be a n by p matrix whose rows Xi,: are all distinct. Show that there exists a vector w of length p such that the scalars (Xw)i are all distinct.

Question 2. Recall the variance of X is Var(X) = E[(X − E[X])2].

1. Let X be a random variable with finite mean. Show Var(X) = E[X2] − E[X]2.
2. Let X and Z be random variables on the same probability space. Show that Var(X) = EZ[Var(X|Z)]+ VarZ(E[X|Z]). (Hint : E[X] = EY [E[X|Y ]].)

Question 3. Let X        be a random variable with density function fX, and g :        be continuously differentiable, where and are subsets of R. Let Y := g(X), which is continuously distributed with density function fY .[pic 5][pic 6]

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