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7%, , , , , SECTION-A [Multiple Choice Questions], Consider a cube C centered atthe origin in R?. The number of invertible Yinear transformations of R? which, map C onto itself is 7, (a) 72 . 48 (c) 24 (d) 12., Consider the following maps from R? to R*;, , * @ the map (x, y) Hy (Qe + Sy + Lex + 3y), :, , (ii) the map (x, ye» (x + 9%, +), and ., Gi) the map given in polar coordinates as(r,0) +» (r,0+r?)for 1 #0, with the origin mapping to the origin., , ‘The number of maps in the above list that preserve areas is: ', (a) 0 (1 (92 @ 3., Consider the vector space V over R of polynomial functions of degree less than or equal to 3 defined on 7, , R. Let T:V + V be defined by (IM) = £2) =f (x). Then the rank of T is, @)1 - m2 °° (3 (4 {NT JAM 2018-MA|, , Let U, V aml JV be finite dimensional real vector spaces, T:U@V, SV Ww and PW —>U be, , linear transformations. Ifrange (ST) = nullspace (P), nullspace (ST) = range (P) and rank (7) = rank (5)., i {UT-JAM 2018-MA], , then which ene of the following is TRUE?, (a) nullity of T = nullity ofS, , (b) dimension of U # dimension of W ., (c) If dimension of V = 3, dimension of U = 4, then P is not identically zero, , {d) If dimension of V.= 4, dimension of U = 3, and T is one-one, then P is identically zero, thet 7:R? > R? be detined by T(4+%2+45) = OH —A2-% =2,.0), IP M{7) and R(T) denote the null space, and range space of T respectively, then . [UT-JAM 2006-MA], (a) dimN(T)=2 — (b) dim RIT)=2— () RTT) ) NTE RLT), , Let S= {7 2K? > R?;7 isa linear transformation with 7(1,0,1) =(1,2,3) and T(1,2,3) = (1,01). Then, Sis : 1 FEE SAM 2008-MA], (a) a singleton set (b).a finite sct containing more than one element, , (c) a countable infinite set (d)an uncountable set, , Let 7: R‘ > R¢ bea linear transformation satisfy 7) 437? =4/ , where / is the identity transtotmation,, then the linear transformation $ = T* 437? -4/ Is, , [UIT-JAM 2009-M Al, , , , =26RS 100K, 2ONH1N09, , New Nethi-16, Ph: 01 ., Dethi-09, Ph: O1L-2765335S, 27654485

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i., , 13., , Y, , Uneer Tronsformations, , {a) one-one but not onto (b) onto but not one-one, , (©) invertible . @)non-invenible, , Let 7:R? > RB? be a linear transformation such that T((1,2)) = (2,3) and T((0,1)) =(1,4). Then, , T(5,6) is : {IT-JAM 2010-MA|, , (a) (6, -1) (b) (-6, 1) (©) (-L, 6) (@) (1, -6), , Let TR? > R® be the linear transfoniiation whose matrix with respect to the standard basis of R’, , "(0 ab 4, , 's}-a 0 c]}, where @, b,c are real numbers not all zero. Then T [IT-JAM 2010-MA}, ab 0 . t, , {a) is ‘one to or (b) is onto a, , {c) does not map any fine passes through the origin onto itself ., , {d) has rank 1, , Let T:R" > R" iaaiiantendliandiie where az2. For ken, let E = (v,,¥,.....%,) 6 R” and, = (Ty,,7¥,....7%,). . "Se, {IIT-JAM 2011-MAJ, , a ‘3, , (a) IPE is linearly independent, then F is linearly independent ,, , (b) If F is linearly independent, then E is linearly independent, (c) If Eis linearly independent, then F is lincarly dependent, (d) If F is linearly independent, then & is lincarly dependent, , For nem let 7; "> R* and T,; "+ R* be a linear transformations such that 7,7, is bejective., , Then . . [UT.JAM 2011-MA], {a) rank (7) = 1 and rank (7,) = or (b) rank (7) = sn and rank (7,) = 0, , {c) rank (7) = » and rank (7,) = 0 (d) rank (7,) = or and rank (7) = wr, , Let IV be a vector space over R and let 7: R* +H’ bea linear transformation such that S = (Te,,Te,,Te,), spans 1° Which one of the following must be TRUE? {UIT-JAM 2012-MA], (a) Sis a basis of IV ‘ “() TUR") 2V, , (c) {Te,,Te,,Te,} spans W @ ker (7) contains more than one clement, , Let P, be the real vector space of all polynomials of degree at most n. Let D:F, > I, and T: P,—> P,,,, be the linear transformations defined by . |IIT-JAM 2013-MA], , Dey + 4,4 0X? + coors +4 (1) 2.0, +208 +t nae, Tay + 8+ 0,2 + coors 0,27) = gx OX) oe eax",, respectively, [fA is the matrix representation of the transformation DT =TD: P, + P, with respect to the, , standard basis of ?,,, then the trace of A is, , @-n (b) ()ntl (ee) :, Let T:R’ +R? be the finéur transformation defined by T(x,y,2)=(x+y,)'+2,2+x) for all, (x,y,2)€R?. Then |NT-JAM 2014-MA), , (a) rank’ (7) = 0, nullity (7) * 3(b) rank (7) = 2, nullity (7) = 1, (c) rank (7)= 1, nullity (7) * 2(d) rank (7) = 3, oullity (7) = 0, , , , Ny ed, , South Delhi : 28-A/11, tin Saral, Newr-l1T Hue Khas, New Dethi-16, Ph : 011-26851008, 26861009, , b North’ Delhi: 33-35, Mall Road, (:T.M1, Nagor (Opp. Metro Gate No, 3), Delhl-09, Mi: 011-27653355, 27654455

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15,, , 16,, , 17., , 18., , 20., , 21., , : * Unear Transformations GD, Let A(R) be the vector space of polynomials inx of degree at most 2 with real coefficients. Let M,(R) be, the vector space of 2 * 2rcal matrices. Ifa linear transformation T: P(R) > M,(R) is defined as, , , , 0)- 0, rine |/ im al [UIT-JAM 2015-MA|, , then, (a) Tis one-«ne but not onto (b) Tis onto but not one-one, , (c) Range (7) na { Ht [7 ; ‘} (d) Null an = span {x* - 2x, t 4, , Let B, = {(1. 2), (2,—1} and B, = {(1, 0), (0, 1D} be ordered bases of R?. 1 TR? R? isa linear trans, formation such that (T']q , the matrix of Twith respect to B, aiid avis = J: 715, 5)is equal wo, , (a) (-9, 8) (b) (9, 8) (c) (+15, -2) (4) (15,2) 1NT-JAM 2015-MA], , Let P be the vector space (over R) of all polynomials of degree < 3 with real coeflicients, Consider the finer, transformation Ts P—> P defined by T(a, + x + a,x" + etx’) = a, + yx + ax? + a,x? Then the matrix, representation Af of T with respect to the ordered basis {I, .r, x, x’) satisfies [IIT-IAM 2016-MA], , (a) M? +1, <0 (b) M?-/,=0 (c) M-1,=0 (d) Af +/,=0, Let P, denote the real vector space ofall polynomials with real coefticients of degree at most 3, Consider the, , map T:P, —» P, given by 7(p(x)) = p"(x)+ r(x). Then [IIT-JAM 2017-MA], (a) T is neither one-one nor onto (b) T if both one-one and onto, , (c) T is one-one but not onto (d) T is onto but not one-one, , Let T:R? + R? be a linear transformation and / be the identify transformation of R?. If there is a scalar, cand a non-zero vector x eR? such that T(x) cr, then rank (7- cf) [IIT-JAM 2005-MA], (a) cannot be 0 * (b) cannot be I (c) cannot be 2 (d) cannot be 3, , Le 7:R’ > R? bea linear transformation such that 71,2, 3) =(1, 2,3), TU, 5,0) =(2, 10, 0) and 7-1,, 2,-1) = (3, 6, —3). The dimension of the vector space spanned by all the eigenvectors of Tis, , (ayo (b) 1 , (c)2 _. 2 @)3 > [NT-JAM 2008-MA], , Let T:R? —» R? be the linear transformation whose matrix: wih respect to standard basis fey, ey gy} of, , oot, ele ' or . [IIT-JAM 2010-MA], 100 . ’ ‘, , (a) maps the subspace spanned by e, and ¢, into itself, (b) has distinct eigenvalues »” . ., (6) has eigenvectors that span R?, , (@) has a mn-zero null space, , a South Delhi : 28-A/11, Jia Sarai, Neat Haws Khas, New Dethi-16, Ph : O11-26851008, 26861009, ESS North Delhi 133-35, Mall Raat, G.T.Ib. Najgwr (Opp. Metr Gate Na. 3), Dethi-09, Ph: O11-27653335,27654455

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: jee Tronsformaticns S&S, ———eeeeeee —=——=, , 22, Let 7:4 R? bethe Tinear transformation defined by, TC 5 %y0%4)) = OLX, — yo Xp 740% THe) ThE which of the following statements are true?, , @ dimKer(7T)=1 if #0 (HCU-2010], (i) dimKer(T) =4 if =D, , Gi) dimKer(T) =1 if Tis onto’, , , *«, , (a) (i) and (ii) , (b) (ii) alone, (©) Gi) and (iii) | (4) (i), (i) and (iii), , 23. Let V be the vector space ofall 2 ¥ 3 real matrices and W be vector space of all 2 « 2 real matrices., ‘Ven = 4 {HCU-2011], ‘@) there is a one-one linear transtwmmation from V —> W ., , tb) Kernel of any linear transformation from VW is nontrivial., ,©) there is an onto linear transformation from V > V, *) there is an isomorphism trom Vow, 24. Let V be a vector space of.dimension i and (¥,5¥p0~ on) baba of Let oe S, and T:V—V, , be a linear transformation defined by T(v,) = Yey)- Then [HCU-2012], (0) Tis nilpotent (b) T is one-one but not onto, (c) Tis onto but not one-one * @)Tisan isomorphism, , 25. Let V be the vector space of polynomials of degree less than or equal to 2. Let D: :V-> V be defined, as Df = f'.1f B,={l.x.x°}, By =(llex’, 1+x+.x7} be two ordered bases, then the matrix of linear, , wansformation [D],. 9, is {HCU-2012], 10 0 o1 0 0 oo, (a) }0 0 -2 h) ]0 o 2 ales" (d)|-2 0 0, 00 2 00 -2 00 2 200, sind §=cos@, , 26. Let Ox Fe R?. For osO0<a, Wt A= ). Ten the ange between F and AF is, , -cos0 sind, , ww 2-0 . fb) O. . (c) =-° (d) 0 {HCU-2014], , 27. ket V bea vector space of dimension 1 over R and {¥j,..00¥,) bea basis of V. Let @ be a permutation, of the numbers (I...) bee 0% HHeccoelt} > (Isat) is 0 bijective map, Then the linear transformation, , defined by 7(¥,) = Yau) Is ‘ . JHCU-2014], {a) J-T but not onto (b) onto but not I-t, (c) neither 1-1 noronto * (d) an isomorphism on V, , 28, LetAbea 7 «5S matrix over having at least 5 linearly, independent rows, Then the dimension of the, null space of A is (HCU-2015], (a) 0 (hb) (c)2 _ (A) at beast 2 1, , , , South Dethl : 2H-A/U, dia Surat, Newr-tlT Mane Khas, New Dethl-16, My: O11-269S100%, 26861009, North Dellel : 33-35, Mull Rawd, GTB, Nagar (Opp, Metre Gate Na. 3), Detht-O9, U's O11-27653355, 27654455

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LUneor Transformations, , 0, 29. Consider.a linear transformation from R‘ to R‘ given by a matrix A= : « Then the number, , con NN, , 0, 0, |, 0, , of linearly independent vectors whose direction is invariant under this transformation is (MCU-2012|, ~ @o wr. (c)2 (@) 4, , = 30. Let V bea 3limensinal vector space over. Let T: VoV be # Bnei iransformation whose characteristic, , payroniys is (X-2X -1(4 +1). Let B be a basis of V. Then which of the following are corrget?, , i 20 0, , (a) The sie of TF wt Bis conjugateto}O -! 0, - ; 001, , ' |HCU-2014], , 1/2, , e 0 0, (b) The matrix of! wet acaba o 1 |, 0 6 -!, , 21 0, (c) The matrix of Tw.ct. B is conjugate to ° -! 1, oo}, , 21 °0, (d) The matrix of Tw.r.. B is conjugate to : -I |, oo}, , 31. Consider the fillowing two statements, $,:There exists a linear transformation T:R’ > R® such that Tis onto and ., , Ker(7) = = (Coates )iM eH +x, =0}, S, : For every linear transformation 7:R? —> R? there exists pe R such that T—yu is invertible, , Which of the following statements are true? {HCU-2015], (a) Both S, und S, are true (b) Both S, and S, are false, (c) S, is false but S, is true (d)S, is true but S, is false, , 32. Suppose 7; P? -» B? is the map T(().%5.45)) 5 (24,.3.2%). Then which of the following is true?, , (a) T has only two distinet eigenvalues , (bd) ker(7") # ((0,0,0)} [HCU-2016), fe) T has three distinet eigenvalues (d) Range of T is isomorphic to R?, , , , eo South Detht 1 28-A/i1, Jia Saral, Newe-t(T Miwz Khas, -New Dethi-16, Mh : O11-26851008, 26861009, E> North Muthl 113-35, Mall Row, GTM, Nagar (Opp. Metro Gate Nu, 3), Dethl-09, My: OF 1-2 7683355, 27684455, , ——,,,,