## Transcribed Text

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 = 2πdet(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.
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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.
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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=14 −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
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