Madurai Kamraj University (MKU) 2007 B.Sc Mathematics maths - Question Paper
(7 pages)
6252/M32
OCTOBER 2007
Paper VII LINEAR ALGEBRA AND NUMBER
SYSTEM
(For those who joined in July 2003 and after)
Time : Three hours Maximum : 100 marks
SECTION A (8 x 5 = 40 marks)
Answer any EIGHT questions.
1. V OT6imjgJ F OT6imD <56TT;5$aiT QgUffiUIT
Q6usrflujiranr(su, MS6mL.eu!i)ef>[D rfjlpeijs.
(<&) a-0 = 0 (ae F, OeV)
(<k) 0-u = 0 (veV, 0e F)
(@) (-a)v = a(-v) = -(av) (ae F,vgV) .
Let V be a vectorspace over a field F . Then prove the following:
(a) a-0 = 0 (ae F, Oe V)
(b) 0v = 0(veV,0zF)
(c) (-a)v = a(-v) =-(av) (ae F ,veV).
2. T :V ->W cresrugi 60 60uiq. 2_0LDrrrr)(DLb erstfld) T<y) = {T(v)/aeV} srarrd <956wni> W -m g_6TTQj0f) gtqst
rfj)|p|6lj<5.
Let T :V >W be a linear transformation. Then prove that the set T(V) = {T(v)/veVj is a subspace of
3. A , B 6T65TU65T ffLD&JlflaOffU-ICfTSTT @0 QfflBJgjgigjl *S46wf)6TT
65TiT6b, At crsffr&jih AB 0 Q<Fraj@655f)
ST65T6L|Lb f)p6tyS5.
Let A and B be orthogonal matrices of the same order. Then prove that A7 is orthogonal and AB is orthogonal.
4. ojiflsnff n &_em_uj @0 <efht asrfl A sresfleb (adjA)A = A(adjA) = |A|7 , srssr I erdsrugji ojiflanff
n &JSKL.UJ 0)6O@c9)685fl.
Let A be any square matrix of order n . Prove that (adjA)A = A(adjA) = \A\l, where I is the unit
matrix of order n .
5. a_6iT Qu0@ Qeusrfl65)UJ 6uan(riup<5. V crssrugji F -sir uSrrssr 6jjiT 2_otQu0@ Qeusrfl 6Tssf]6b
(*s) (u, av) = a(u,v) ae F,u,veV
("4b) (u,v+w) = (u,v) + (u,w), U,V,W&V CT65T
Define inner product space. If V is an inner product space over F , prove that
(a) (u, av) = a(u,v), ae F,u,veV
(b) (u, v+ w) = (u,v) + (u,w), where u,u,weV .
<>rfT(D 655flu51r iljDuiSlujaiq
8-6 2 -6 7 -4 2-4 3
A =
ffLD65TUnL.6ff)L_flj 6fT6Wr5j.
Find the characteristic equation of the matrix 8-62'
-6 7 -4
A =
2-4 3
T6ST(D QfflEJQjSgjj
14 3 2
1 2 3 4
2 6 7 5
7.
6j)iDui51fb@<5 @5)|d) arrasra.
'1 4 3 2 12 3 4
to normal form
Reduce the matrix
2 6 7 5
and find its rank.
8. N otottjd OTassrasiflair, sy@uurr6T6tfla5r OT6WTSsrf)ffi65>ffi0nujs ffirrcwTiK.
Find the numbers of divisors of a given number N.
9. n(n + l)(2n+l) cramo ctw, 6 j,d) 6u@uld srasi rf|)fp6q.
Show that n(n +1)(2n +1) is divisible by 6.
10. 21000 =3(modl3) GT6Tlp6i|.
Prove that 21000 = 3 (mod 13).
4 6252/M32
11. 9 GPCtP OTSTOT655fl6irr dJirffiffiLD 3m ebeog] 3m+ 1 crayip euii|_a51d) |0@lc) otsst<5 asml*.
Show that every square is of the form 3m or 3m +1.
12. 7121+1 ST6STp 6r6ror 719-cb cu@uLb OT65T Show that 7121+1 is divisible by 719.
SECTION B (6 x 10 = 60 marks)
Answer any SIX questions.
13. 90 (LpuGrTCfT uifliorrsror Qeu<si_ir QcuGifluSlsin g-Ggjiih
rffcOOl Q) c|lfj_5jfl365ffTnij<9j6TT ffLD CT65ffT55cf]i$lD$LL(GiTTT (o)QJ<5l_IT<S65)CTTLJ
QurbrSl0ffi0ii> OT65T
Prove that any two bases of a finite dimensional vector space have the same number of vectors.
14. <s6mi> F -6sr lS65)ld Qcu.$i_it Qeusrfl V -<sirr s-en-Qeustfl W crssfleb, ~ = {W + V7veV} j,65Tgi F -ot lS)
90 Qajffiuir Qeuerfl crafT j)p64<s.
If W is a subspace of a vector space V over the
field F , prove that = {W + V7yeV}isa vector space W
over F .
15. OTrijaj ffgjij 65sflnujiL|ii), 0 ffLDffiT ssrfl n>fi)pii> 6it OTdlir r cSiaiBfl <j,liu6uiT)j51a!T a.(la;6Drr<5 6j&G|7 6p0 GuySluSlo)
CT(Lp (LplT}-U_(Lb OT65T lp6I|ffi.
Prove that any square matrix is expressible uniquely as a sum of a symmetric matrix and a skew symmetric matrix.
16. eijQeutr (LpLq_<si_|<sh'<srr uifliBirsrorcnOT 2_gttQlj0.$0 QcuifltL|Lb, 60@ U)_6)!)ujlj QufT)i5l0ffiLb srayr
Prove that every finite dimensional inner product space has an orthonormal basis.
17. QffiujaSl-ajDn-uSlcbuOTr cr(Lpl iglpajffi.
State and prove Cayley Hamilton theorem.
18. 6eijQcurr0 sh-L. crssffranasOTiLiih, u<srr er6ror)Seifl6jr Qu0ffiffi6O(T 6G[T 6p0 61JLluSleb er(Lp (Lpiq_U_JLD 6T65T (0p6I(ffi.
Prove that every composite number can be written as a product of prime numbers in a unique way.
19. n fi0 (y)(ip cresTr srasfld), <p{N) -ssr ld<$ui_| <Knwr<s.
If n is an integer, find the value of <p(N).
20. (s\) 34n+2 + g2n+1 OT65Tp otsrst, 14 ,0) eu@u(5\ii> otqst
flp<51|(5.
(b) X, y, Z OTCTTU65T (ipOTp (Lp(Lp OT651Sr.OT
ersiflsb, (Z*)3 -3Z;e3 eraiTugj 108 <,60 eu@uLb crara
<[T65ffr<S.
(a) Prove that 34n+2 + 52n+1 is divisible by 14.
(b) If x, y, z are three consecutive integers show that (Zxf -3Z*3 is divisible by 108.
21. Qeo<5(r(r(S5#laSlk G&, (bpis&ns) Gr(Lpl Slamor GgprDf&mps, 0efil(S.
State and prove Lagrange's theorem. Deduce Wilson's theorem.
22. x + y + z= 6, x + 2y + 3z=14, x + 4y + 7z=30 ergyii) ffLDeirrurr(5<5sn' Qurr0rf)gjLb srm& ffimlii). ireLjaaoerT.s iS n"ioffffT<5.
Show that the equations x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and solve them.
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