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Unformatted text preview: DO THREE OUT OF FOUR LINEAR ALGEBRA PROBLEMS Linear Algebra 1. Consider the systoni of linear equations: :1: «~ y 4* k 3 i 0
.z' A: y A: i (1
A: .1' W y + z : b (a) For what values of the constants A: a. and c will the syslmn liaw a unique
solution: (ii) multiple solutions; (iii) no solution'.2 (b) \Vlion there is a uniquv solution, What is it? (t) \Vlion then! are 111tllllplt‘ solutions‘ find the (liniension and a basis of lll(‘ solution
space to the associated lioinogcnons systeni. 2. Consider the upper triangular nnitrix A. defined by: 1 a I)
A : U 1 (1
0 0 l (a) Find an expression for A”. where 71, is any positive integer.
(1)) Find an expression for A”. where n is any negative integer, (0) Comment; on the, meaning of Aoi 3. (Tanwidor the subspace of R“ deﬁned by \v' : 51)a11(S), whoro: 5:{ (a 0,5 0).
(1 1.11)‘
(1 71.1 1),
(2 2. 2 A1 )} (elf) \Vhen is Ihv (hlllffllSiOH of 1”.
(1)) Find an 01'1‘1101101‘111111 basis for \V. (t) [5 the Vector (1:13:41) in V.) l. (liven IllO linear lI‘zlllsl'ol‘lllill‘loni I“. on R3 deﬁned by:
F(:r. (I/i : (3:1‘ 7 yi ~.I‘ *7" 2g 7 3‘ 7y + 3:) (A) Find all eigonvalncs of the transformation, and 21 basis for each associated cigcnspeu'e.
ls F (li2100112Llizzil)le'?
D (1)) Find A transfornizilion G. such that C O (I : F. D0 THREE OUT OF FOUR ADVANCED CALCULUS PROBL.“‘.MS Advanced Calculus
, . . 9 ,
l. Consulcr lilIC (‘111'\'(‘, in R“ Llelinod by: HT) : :1“ + 2.144 .17
.1 ‘ (it) Find any points of discontinuity and the lllllllb' of in the neighborhood of
111950 point‘bx as well as the lllllllh‘ as .l' —> :00 (b) Find the points corresponding to all looks. niuxiinzi. niininni and inflec
tion points of. (ﬂ Skolcli tlio curve. \\'11211 are 1119 bonndw if any on ﬂﬂz”)? ‘2. Consider [11v function : :1" si11(\:1~ i 71'). (21) \ViizLL is the total ‘(LI'QEL lying between [1110 X~2LXiS£1lld fﬂl‘) 011 11110 doiiieiiii (*7. +7)? (1)) \Vlmb is 1110 \'01111110 of 1110 solid fanned by romting this region around 11110 yrnxis
(0111‘ full 1'm'o1111i011)? The plane and cylinder in R3 defined by:
.'I'*Llj+l:l 21ml:
,1'2 + g/2 : 1 intersect in a space curve Find the 11121Xi111L1111 and 111i11111111111 \ullrles 0f the function: y‘ 2) :117+ 22/ + 3: 011 this curve, "1 Consider the function
[(lx y, 2) :1'2 +1y2 + 9:") Using an appropriate coordinate substitution‘ ﬁnd the value of: / // €<f(.r..z/.:>>5(ﬂx
 R where I? is defined by f(1]‘7‘l/,Z> <: 16. ...
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 Fall '08
 Feinberg,E

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