Consider the case where X is an n x 2 data matrix, containing the values of two variables and n is the number of observations. Both variables have mean 0 and variance 1. Let S be the variance covariance matrix and w be the eigenvector of S corresponding to the largest eigenvalue. Let v be any arbitrary 2 x 1 vector with a length of 1. 1. What will be the dimensions of w? (1 Mark) 2. What will be the dimensions of a where a = Xw? (1 Mark) 3. Describe what is contained in a. What do we call this? (1 Mark) 4. Let the value of New Variable A for observation : be given by a, = with + war,2. Let the value of New Variable B for observation i be given by by = v1In1 + 27/2. For clarity, wj and w2 are the first and second elements of w. v, and v, are the first and second elements of v, r, is the element in row i and the first column of X and r,2 is the element in row i and the second column of X. Write down an expression for the sample correlation between New Variable A and New Variable B in terms of w, v and S only? You may use the result that (FG)' = G'F (3 Marks) 5. What does it mean for w and v to be orthogonal? Prove that New Variable A and New Variable B are uncorrelated only if w and v are orthogonal. (3 Marks) 6. Suppose there is a matrix with w as its first column and v as its second column and w and v are orthogonal. What type of matrix is this? (1 Mark)