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Pipscuadeny, ‘An [SO 9001 : 2008 Certified institute, , , , , Linear Transformation and Its Pro}, , , , 1: Let V be the space of twice differentiable, functions on R satisfying f"-2/'+ f=0., Define T:V +R? by T(f)=(F'(0),£(0))., Then 7 is, , (a.) one-to-one and onto., , (b.) one-to-one but not onto., , (c.) onto but not one-to-one., , (d.) neither one-to-one nor onto., , Given a 4x4 real matrix A , let, T:R‘—R® be the linear transformation, , defined by Tv = Av, where we think of R*, as the set of real 4x1 matrices. For which, choices of A given below, do Image(T) and, , Image (7?) have respective dimensions 2, , and 1? (* denotes a nonzero entry) ~ -, , , , , foo * *], 0oo* *, @)4=)) go +, lo 0 0 0], fo o * oO], 00*0, b.) A= ., (b) 000 * So, lo oo *|, [0 0 0 Oo], 0000, cc.) A=, (4-1) go *, lo o * o|, [0 0 0 Oo], 0000, (GC) AS! pe», 00* *, 3. Which of the following is a linear, , transformation from R? to R??, , x, , 4, (IE), +, z ety,, x, xy, «Ce, +, z may,, x, , , , , , , , , , , , z-x, o Na (33), , , , , , ASSIGNMENT-2, , , , z, # 7: 9999TBIAS-4 & 9EIIIGITI4, SSESEMTEY, ‘Near LLT., New Delh}-110016, Ph: (011}-26537527, Cell:, Le I, Fat Meee) J Sarah ae Dae Tfocodipsacademy.com: Website: www dipsacademy.com, , (a.) only f., (b.) only g., (c.) only A., (d.) all the transformations f,g and h., , 4. Consider non-zero vector spaces Vj,V2,V3,V4, and linear transformations $,:V,->V>,, $2:¥2 V3, $3:¥3->V%_ such that, , ker(,)={0}, Range ($;)=ker(2), Range, ($2)=ker($3), Range ($;)=V4. Then, , a 2, (a.). >(-1)' dimy, =0., i=l $, , “S_) ¥(-1!dimy;>0., , #=2, , 4, (c.) ¥(-1)'dimy, <0., =I, , 4, (4.) >i(-1)' dimy, #0., f=l, , Let M,(K) denote the space of all nxn, matrices with entries from afield K . Fix a, non-singular matrix A=(4y)EM,(K) and, consider the linear map, T:M,(K)— M,(K) given by: T(X)= AX., Then, , (a.) Trace (7) = a An, (b.) Trace (T)= Pi Lier Ay, , (c.) Rank of T is n?, (d.) 7 is non-singular, 6. Let Mmxn(R) be the set of all mxn, , matrices with real entries. Which of the, following statement is correct?, , (a.) There exists 4 €M2,5(R) such that the, dimension of the null space of A is 2., , (b.) There exists A€M,5(R) such that the, dimension of the null space of A is 0., , (c.) There exists AeM2,5(R) and, Be Msy9(R) such that AB is the 2x2, identity matrix, , (d.) There exists A€ Mz.5(R) whose null, , z 5., space is {Car 2x5 2445) € R°:x

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=, Lr ., An 1S0 9001 : 2008 Certified Institute, , , , 10., , 21g =H Xg = Ny }, , Let ¥ be the vector space of polynomials, over R of degree less than or equal to 7. For, define a, T:iV73V by, (Tp) (x) = a9 - a,x + ax? —....4(-1)"ayx", Then which of the following are correct?, , (a.) T is one-to-one, , (b.) T is onto, , (c.) T is invertible, (d.) det T=0, A_ linear, , P(x) =agtayxt...ta,x" inV,, , linear —_ transformation, , transformation T rotates cach, , vector in R? clockwise through 90°. The, matrix 7 relative to the standard ordered, , =, , (a.) [S a J, , of], (©) k al, wf 3], , Let 7:R” — R” be a linear transformation., , Which of the following statements implies, that 7 is bijective?, , (a.) Nullity (7)=n, , (b.) Rank (7) = Nullity (T)=n, , (c.) Rank (7) + Nullity (T)=n, , (d.) Rank (7) -Nullity (7)=n, , Let n be a positive integer and let M,,(R), denote the space of all mx» real matrices. If, T:M,(R)3M,(R) is a ___linear, transformation such that 7(4)=0 whenever, , AéeM,(R) is symmetric or, symmetric, then the rank of 7 is, , n(n+1), eae Td, , skew, n(n-1), (b.) =>, , , , Rte ROMO mtd a, , (c.)n, (d.) 0, , 1. Let S:R¥-> Rt and T:R4 +R? be tincgr, transformations such that 7°S is the identity, map of R?. Then :, , (a.) S°T is the identity map of R4, (b.) S°T is one-one, but not onto, (c.) S°T is onto, but not one-one, , SK ‘a S°T is neither one-one nor onto., , @~ ‘Let a,b,c,d eR and let T:R? > R? be the, SS Minear defined by, , . r((; }) = [= = ra for ["] eR? , Let, y ex+dy y, , S5:C—C be the corresponding map defined, , transformation, , , , = by S(x+iy)=(ax+by)+i(cx+dy) — for, : x,yeR. Then, , (a.) S is always C_ -linear, that is, , S(z+22)=5(z)+S(z2) for alll, , 2,22¢€C and S(az)=aS(z) for all, aeCand zeC., , (b.) S is C-linear if b=-c and d=a, (c.) S is C-linear only if b=-c and d=a, , (d.) S is C -linear if and only if 7 is the, identity transformation., , 13. Let n be a positive integer and V be an, (n+1)-dimensional vector space over R.. If, , : ‘, y, {e1,Cas4€nu} is a basis of } and, , TVOV is transformation, , the linear, , satisfying T(e)=ej4) for f=1, 2,00! and, T(€n41)=0. Then, , (a.) Trace of T is non zero, , (b.) Rank of T is n, , (c.) Nullity of 7 is 1, , , , 28A/11, (First Floor) Jis Saral, Hawz Khas, Near LUT. New Dethi-110016, Pht (011)-26537527,, E-mall: jafo@.diptacademy.com: Webslie: www. dipsscade:, , , , , , , , Cell: 9999183434 & 9899161734, 8588844789

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14., , , , , , _ a wv - —_, Pebescciens, , ‘An 1SO 9001 : 2008 Certified Institute, , 15., , ee x, , (d.) T” =ToTo,..0T (ntimes) is the zero, , map., , Let ” be the vector space of all real, polynomials of degree at most 3, Define, SVOV by, S(p(x)) = O(x), Vp(x) EV, Where Q(x) = p(x +1),, , Then the matrix of S in the basis, , {.x, xx}, considered as column vectors,, is given by:, , , , 1000, @|° 29°, , “10 030, , loo 0 4, , Prauid \, 0123, 5013, , loool, , fl 123 ., 1123, ©), 223, , 133.3 3 ~, fo0 00, , 1000, , d., @lo 100, , lo 0 1 0, , For a positive integer n, let P,, denote the, space of all polynomials p(x) with, Coefficients in R such that deg p(x)<n,, and let B, denote the standard basis of P., given by By, = {Ixy x2, sux} . We, T:P;— Py is the linear transformation, , defined by T(p(x))=3?p'(x)+ ff ple, and A=[ay] is the Sx4 matrix of T with, , Fespect to standard bases By and By, then, , (1.) ayy =F and ay; -1, , (b.) ayy = 5 and ay; =0, , 16., , Q:, Y, , 17., , 18,, , Linear Transformation and [ts Properties, 7, , (c.) ay =Oand a3 =y, , (d.) ay2 = Oand ay; =0, , Consider the linear, T:R’ +R’ defined by, , T(x ye gees 087) = (876 G0 0-08 870M), Which of the following statements are true?, (a.) The determinant of T is |, , (b.) There is a basis of R? with Tespect to, which T is a diagonal matrix, , (c.) T?=1, , (d.) The smallest n such that 7” = / is even, , Let W be the vector space of all real, polynomials of degree at most 3. Define, T:W—W by (T(p(x)) = p(x) where p' ts, the derivative of p. The matrix of T im the, , basis {1, * ex}, considered as column, vectors, is given by, , 000 0), , transformation, , (a.), , (b.), , ounoo oo, , (c.), , co- ooo, (a), , opoeocooeoeoHe-s eo, couvowososo0o0owss, , 0, 00 0, Let x, be linearly independent vectors m, R* suppose T:Ri +R? is a hnear, transformation such that Ty = arand Tr =O., , , , , , coon ooNoHwoOD oO ONO, , Then with respect to some basis in R°\T of, the form, , a.) £ *\.a>0, a 0 al, , (b.) (3 P)absQars, , , , , , , , , Near LUT. New Deth- 110016, PR: (OUIP2ESI7S27, Cot PUOOLATAM A GEPUTOLTM, ESERBALTED, Al; Info@dipsacademy.connt Website: www.dipancademy.com

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[, , 19., , 20., , 21., , 22., , 23., , , , () 0 a, “lo 0, , 0 0, wl), , Let V be the space of all, , transformations from R? to R? under usual, addition and scalar multiplication. Then, , linear, , (a.) V is a vector space of dimension 5, (b.) V is a vector space of dimension 6, , (c.) V is a vector of dimension 8, , (d.) V is a vector space of dimension 9, , Let A:R®-»R> and B:R53R’ be two, , linear transformations. Then which of the ~ Se, ‘ ~(d.) None these, , following can be truc? ., , (a.) Aand B are none one-one, *~, , (b.) Ais one-one and B is not one-one = Ss, (c.) A is onto and B is one-one KY N~, Tv, (d.) A and B both are onto. \, The transformation», (x. 9.2) (x+y. y4+2):R’ > R? is ot al, (a.) Linear and has zero kernel, (b.) Linear and has a proper subspace as 26., kernel, (c.) Neither linear nor 1-1, (d.) Neither linear nor onto, Let T:R> + W be the orthogonal projection, of R’ onto the x plane W’ . Then, (a.) T(x, yz) =(x+y, 0) y+z), (b.) T(x, »,2) =(x-y,.0, y-2), (c.) T(x, y,z)=(x+y+z,0,2) 27., , (d.) T(x, y,z) =(x,0,=, , {v,.¥.¥,) is a basis of V=R? R’ and o, linear transformation T:V — V is defined by, T(vJemrreT(vs)=¥, 4¥.T(Yy Jey, +y,,, , then, (a.) Tis 1-1, , 2SA/11, (First Floor) dia bared, Haus Khas, Near LT. New Delld-1}0016, Fis (017, E-mail: infuit.dipenc adem sum Website:, , ¥20897811, Cott: 9999105454 & 9099161734, BSREUIITED, , An ISO 9001 : 2008 Certified Institute, , (b.) T(y +0 4+ v,)=0, (c.) T(v, +2v, +4) = 7(v, +24), (d.) T(y-y)ay ny, , For the standard basis, * {(1,0,0),(0,1,0),(0,0,1)} of RB’, a linear, , transformation 7 from R’ to R? has the, , 2 =, matrix representation | 1 | , The, 3 1 -2, image under 7 of (2, 1, 2) is, (a) (11, 0, -1), ~(b.) (11, 3, -5) {c.) (7, 3, -1), , For a linear transformation T:R" + R‘, the, , kernel is having dimension 5. Then the, dimension of the range of T is, , (a.) Five, (b.) Six, , (c.) Four, (d.) Two, , 1-1 0, Consider the 3x3 matrix ri -2 )., ol -l, , Which of the following is false?, , (a.) There is a non-zero vector which is not in, the image of T, , (b.) There is a non-zero vector which is in the, kernel of T, , (c.) There is a non-zero vector which is not in, the kernel of 7, , (d.) The matrix is invertible, , Let ¥ be the vector space of 2x2 matrices, over R. Which of the following is/are not, linear transformations?, , A. T((4,)) =a,, , T(A)= A+], , B., C. T(A)@= Trace A, D., , T([a, Je ant ay tay tay, , pp, , , , ww dipenc ad:

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18., , 29,, , 30,, , , , , , X, , (d.) Only D, , be the real vector space of real, olynomials of degree < 3 and let T:V +¥, me the linear transformation defined by P(t), and O() where Q(1)= p(at+b). Then the, matrix of T with respect to the basis 1,¢,/7, of V is:, , 1b, , @)|9 4, 0, , Let ¥, , (b)}0 & 2ab, , (.)|a a 0, , aqaaa@, dj} b 6 0, a B, , Let T: IR —>R4 be a linear map defined by, T(x,y,z,w) =(x+2,2x+ y+32.2y+ 22.0)., Then the rank of 7 is equal to, , Let 7),7> :R5 — R? be linear transformations, such that rank (7})=3 and nullity (7;)=3., Let 7; :R? — R? be a linear transformation, such that 7,°7%)=7. then rank (Tj) #8, , , , , , ——_, , Let Mf be the real vector space of 2 * 3, matrices with real entries, Let T: MM be, , wf 2 2), y Xy is %, , [3 “I |: Then determinant of T is, , defined, , My oxs xy, _—_—————‘ : 2 R), Let linear transformation T:R?->R™ be, ). Then, , defined by T(x,, , 2) = (yy HDD, the nullity of 7 i, , , , mall: tafadtaliosas oem ©”, , aes et A CPT na Lathan tad, , uw, ‘WU (teat Floor) dla Sarak, Hawa Khan Neer LET ent, , , , , , , , , , (a0, (b.)1, (c.)2, (d)3, Let Fy =(plx)! p(x) be a polynomial with, real coefficrent and degree at-most 3) end, T:R+P be the map prea by, , , , , , , , , , , , , , , , , , , , , , , , , , T(pix))=] pnd If the mtr of T, ', relative to the standard «= bases, , Rsh=(Ler.r} s M ad we, , denotes the transpose of the matrm Af . then, M+M' ts, , nag a, , 1, oon, a], , 1, >, °, , a, woo, ', tw, co oM, ——__, , onws, , -1), 0, -1, , l 0, fo 2 2], 2-1 0}, 20 -1 0), , o-—-no, w- so, , mah, , , , o 0 -u, fet 2), If the nullity of the maar | 1 =! -liel, tyr 4], then the value of ¢ 6, (a)-l, (hyo, «cl, qad)2