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1. (a) (b) (c) (d) (e) The matrix A has an eigenvector w as follows:  3 1−1 1 A= 2−3 4 and w=1 −1−1 5 1 Show that w is an eigenvector of A and find the corresponding eigenvalue. Find the eigenvalues and eigenvectors of the following matrix 􏰀9 2􏰁 V=26 Use your answer from part (a) to find the principal axes of the following ellipse 9x21 + 4x1x2 + 6x2 = 20 Sketch the ellipse in the original x1, x2 coordinate system, and mark the principal axes of the ellipse on your sketch. A two dimensional normal distribution has the probability density function (p.d.f.) 1 1 −(xT V −1x)/2 f = 2π􏰃det(V)e Using your answer to part (b) give a change of variables which separates the probability density function f into the product of two single variable p.d.f.s. 1 3. A probability density function fT(x,y) for a two dimensional random vector T=(X,Y)T hastheform 􏰂kx(1−y)(y−x) x≥0, y≥x, and y≤1 fT (x, y) = 0 otherwise (a) Sketch the region D on which fT is non-zero. (b) Using a double integral, calculate the value of k which makes fT a proba- bility density function. (c) The p.d.f. fT has a single stationary point inside the region satisfying x > 0, y > x, and y < 1. Find this stationary point. 2 4. (a) Calculate the determinant of the matrix (b) 031 A=1 0 1 235 by reducing it to row echelon form. The matrix B has a row echelon form R, where 120212 120212  −1 1 3 0 0 −2   0 3 3 2 1 0  B= 1−4 −6−2 0 3 and R=000011   −3 3 9 0 −2 −2   0 0 0 0 0 6  6 −6 −18 0 3 3 0 0 0 0 0 0 i. What is the rank of B? ii. Find a basis for the column space (also known as the range space) of B. iii. Find a basis for the null space of B. Data points x1, . . . , xN in R2 have mean μ and variance matrix V as follows (c) (d) (e) Ifnewdatapointsz1,...,zN areconstructedviathelineartransformation zi = Gxi, i = 1,...,N, find the mean μz and variance matrix Vz of the zi data in terms of μ and V when G=1􏰀4 −3􏰁 534 Show that G is an orthonormal matrix (the term orthogonal matrix is also used). Without performing any numerical calculations, how are the eigenvalues of V and Vz related? Justify your answer without using any numerical calculations. μ= 􏰀 5􏰁 −10 and V= 􏰀2−1􏰁 −1 3 3

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