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20
84
mRrj iznsk jktfkZ V.Mu eqDr fo'ofo|ky;] bykgkckn
vf/kU;kl (Assignment) l= 2012&13
Lukrd dyk dk;ZØe
Under Graduate Art Programme
fo’k; % xf.kr
Subject : Mathematics
dkslZ “kh’kZd % Linear Algebra
Course Title : Linear Algebra
Section - A
[k.M & d
fo’k; dksM% ;w0th0,e0,e0
Subject Code : UGMM
dkslZ dksM % ;w0th0,e0,e0-02
Course Code : UGMM-02
vf/kdre vad % 30
Maximum Marks:30
vf/kdre vad % 18
Max. Marks: 18
uksV % nhkZ mRrjh; iz”uA vius iz”uksa ds mRrj 800 ls 1000 “kCnksa esa fy[ksaA lHkh
iz”u vfuok;Z gSA
Note : Long Answer Questions. Answer should be given in 800 to 1000 words.
Answer all questions.
1 Define a vector space and given an example of it. Docs there exist a finite
vector space over an infinite field 6
osDVj Lisl dks crk;s rFkk ,d mnkgj.k nsaA D;k vuUr QhYM esa ,d QkbukbV
osDVj Lisl gksrk gSA
2 Define the rank and nullity of a linear transformation.
Verify that the map. 6
2 3
2
T : R ® R given by T ( x1 , x2
) = ( x1
+ x , x1
- x2
, x2
) is linear. Find
the rank and nullity of T.
2 3
js[kh; VªWlQkesZ'ku dk U;wfyVh ,oa jSad crk;sA crk;s fd eSi T : R ® R
2
given by T ( x1 , x2
) = ( x1
+ x , x1
- x2
, x2
) js[kh; gSA T dk U;wfyVh ,oa
jSad crk;saA
3 Reduce the following matrix to the normal form and hence find its rank.
fuEu eSfVªDl dks ukeZy QkeZ esa fjM~;wl djrs gq;s bldh jSad crk;sA
é1
2 -1
2 1 ù
A =
ê
ú
ê
2 4 1 - 2 3
6
ú
êë
3 6 2 - 6 5úû
Section - B
[k.M & [k
vf/kdre vad % 12
Max. Marks: 12
uksV % ykq mRrjh; iz”uA vius iz”uksa ds mRrj 200 ls 300 “kCnksa esa fy[ksaA lHkh
iz”u vfuok;Z gSA
Note : Short Answer Questions. Answer should be given in 200 to 300 words.
Answer all questions.
4. Let V be a vector space of dimension 100. If the dimensions of the sub
spaces W 1 and W 2 of V are 60 and 63 respectively find the maximum and
minimum dimensions of W1 Ç W2
3
ekuk fd V ,d osDVj Lis'k gS ftldh foek 100 ;fn nks miLis'k W 1 rFkk W 2
dh foek 60 rFkk 63 gks rks W1 Ç W2
dh U;wure rFkk vf/kdre foek Kkr
djsaA
5- Let V and U be vector spaces over the same field F. prove that the linear
Transformation. T : V ® U is injective if and only if Ker T = { 0}
3
;fn V rFkk U nks osDVj Lisl ns rks fl) djsa fd js[kh; VªkUlQkeZ
T V ® U
Ker T = 0
: esa ,dkadh gksxh ;fn vkSj dsoy ;fn { }
6 By Cayley-Hamilton theorem, find the inverse of the matrix
dSyh gSfeYVu izes; ls fuEu eSfVªDl dk inverse Kkr djsa