Note: A number of you requeted solutions to the sample exams on the course web page. I presume these answers are correct, but I have done them quickly and may have made errors. I have not cleaned up the questions themselves and I have not put in all the steps in the answers. -------------------- Test 1 5/22/2000 1. (20) Let A b e an y 3 ? 3 matrix with eige nvalu es 1, ?2, and 3. Answ er th e follo wing, givi ng adequ ate reasons. a) Wh at are th e eige nvalues of A T ? b) Do es A? 1 nece ss ar ily exist? If it do es, what are the eigen valu es of A?1 ? c) What ar e the eige nvalu es of A 2 ? d) Is A nece ss ar ily di agonali zab le? e) Is A nece ss ar ily orth ogonal ly di agonali zab le ? a) A^T has the same eigenvalues as A, 1, -2, and 3. b) Since 0 is not an eigenvalue of A, we know null(A) = {0} so A^{-1} exists. The eigenvalues of A^{-1} are 1, -1/2, and 1/3. c) 1, 4, and 9 d) A must be diagonalizable because there are 3 distinct eigenvectors. e) A might not be orthoganally diagonalizable, since it might not be symmetric. 2. (20) Let u1 , u2 , u3 , u4 b e an or thon ormal set of vec tor s in a four dime nsional in ner pr o du ct space V . a) Wh at is || 2u1 + 3u2 ? u3 || ? b) Sh ow that u1 ? 2u3 + u4 and 2u1 ? u2 + u3 are orth ogonal. c) Is u1 , u2 , u3 , u4 a lin ear ly ind ep end ent li st of vec tor s? d) Is u1 , u2 , u3 , u4 a basis of V ? a) sqrt(2^2+3^2+(-1)^2) = sqrt(14) b) = 1x2 + 0x-1 + -2x1 + 1x0 = 0. So they are orthogonal. c) Yes, any orthonormal set is lin ind. d) Yes it is a basis since it is 4 lin ind vectors in a 4 dim space. 3. (20) Find an orthogon al bas is for P2 with inn er pr o du ct < p, q >= ?1 p(t)q (t) dt. Next fin d an or thon ormal basis for P2 (with the same inn er pro duct). Use gram-Schmidt. Start with a basis, say 1,t,t^2. Let u1 = 1, u2 = t - /<1,1> 1 = t, u3 = t^2 - /<1,1> 1 - / t = t^2-1/3. then u1, u2, u3 is an orthogonal basis. An orthonormal basis is obtained by normalizing, 1/sqrt 2, sqrt(3/2) t, sqrt(5/8)(3t^2-1) 4. (15) Let T : P2 ? R 3 b e th e lin ear transfor mation T (p) = [p(0) p(1) ? p(?1) p(2)] T . Fin d th e matrix for T relati ve to the basis {t 2 , t, 1} of P 2 and {e1 , e2 , e3 } of R 3 . T(t^2) = [ 0 0 4]^T, T(t) = [0 2 2]^T, T(1) = [1 0 1]^T so the matrix has those columns in order: [0 0 1; 0 2 0; 4 2 1] 5. (26) Le t x1 and x2 b e two diffe rent leas t squar es solut ions to A öx = b for a 7 ? 3 matr ix A. (S o x1 = x2 ). Cou ld th e ran k of A b e 3? Wh at can you say ab out Ax 1 ? Ax 2 ? Wh at can you sa y ab out Ax 1 ? b? Giv e suffic ient reas on s for your answ ers. Fin d the leas t squares soluti on to ? ? 1 1 0 1 0 1 ? ? x y = ? ? 1 2 3 ? ? The rank could not be 3, since then there would be only one least squares solution, (since A^TA would be nonsingular). But Ax1-Ax2 = 0 since Ax1 and Ax2 are the closest point b-hat to b in the column space of A. Ax1-b is perpendicular to the column space of A. The normal equation is [1 1; 1 3] [x; y] = [1; 6] which has solution y = 5/2, x = -3/2. So the least squares solution is [-3/2; 5/2]. 6. (25) Fin d an orth ogonal co ord inat e chan ge matrix P whi ch tran sforms th e quad rati c for m x 2 + xy + y 2 + z 2 into on e with no cross pro duct te rms. The associated matrix is [1 .5 0; .5 1 0; 0 0 1] with eigenvalues 1, 1/2, and 3/2 and eigenvectors [0 0 1]^T, [1 -1 0]^T and [1 1 0]^T normalizing these to get an orthonormal basis and making them the columns of P we get P = [ 0 1/sqrt2 1/sqrt2; 0 -1/sqrt2 1/sqrt2; 1 0 0]. 7. Not covered in our class. 8. (10) Gi ve an exp licit example of the sin gul ar valu e decom p osition of a 3 ? 4 matrix with rank 2. (You nee d not mul tipl y it out). The singular value decomposition in the case would be USV^T where U is 3x3 orthogonal, V is 4x4 orthogonal, and S has an upper left 2x2 diagonal with positive diagonal entries and the rest of S is 0. An easy example is U = I_3 and V=I_4 the identities and S = [2 0 0 0; 0 1 0 0; 0 0 0 0] 9. (54) Tru e (alw ays tr ue), False (alw ays false ), Ma yb e (some ti mes tr ue an d som etimes fal se, dep en din g on A, S , etc.) or short an swer. a) If the characte ristic p ol ynomial of A is ?(? ? 3)( ? + 3)( ? ? 10) th en A is di agonali zab le. b) Two eigen vec tor s of a symme tr ic matrix are if th ey cor res p ond to diffe rent eige nval ues . c) If A is invertib le, then A is row equi val ent to th e id entit y matr ix. d) If A is similar to B and A is sin gular , then B is sin gular . e) If L: R 9 ? R9 is a lin ear tran sformation an d the kern el of L is {0}, then th e range of L is al l of R 9 . f ) If x an d y are two vec tor s with the same length in an inn er pro duct spac e, then (x ? y ) ? (x + y ). g) An y squar e mat rix A can b e wri tte n as A = S J S ? 1 where J is in Jor dan canon ical for m. h) For A, B , and C matrices , if AB = AC th en B = C . i) Th e eige nval ues of a symme tric matr ix are al l real . j) det( B A) = det( AB ). k) If A is an n ? n matrix , th en A is diagon alizable if R n has a bas is of eigen vectors of A. l) If A has eige nvalu es 1,2 and 3 th en A + 3I has eige nvalu es . m) C ol (A)? = . n) If S is a sub spac e of an in ner pro du ct space V , th en define S ? . o) If A is an m ? n matrix with ran k n, th en th e matr ix of or thogon al pro jec ti on to C ol (A) is . p) If n vec tor s span an m di mension al spac e, then i) n ł m ii) n ˛ m iii ) n = m iv) You cou ld some ti mes have n ł m and som etime s have n < m, dep endin g on the vec tor s. q) If there ar e n lin early in dep end en t vectors in an m di mension al spac e, then i) n ł m ii) n ˛ m iii ) n = m iv) You cou ld some ti mes have n ł m and som etime s have n < m, dep endin g on the vec tor s. r) Ax á y = x á A T y . a) T b) linearly independent c) T d) T e) T f) T g) T (but not covered in class) h) S i) T j) T k) T l) 4, 5, 6 m) Nul(A^T) n) S-perp is the subspace of all vectors v so = 0 for all w in S. o) A(A^TA)^{-1}A^T p) i q) ii r) T ------------------------------------------- Test 2 5/22/2000 1. (25) Let A b e a 3 ? 3 symm etri c matr ix with ran k 2 an d supp os e A ? ? 1 1 ? ? = ? ? 1 1 ? ? and A ? ? 1 ?1 0 ? ? = ? ? ?1 1 0 ? ?. Wh at are all the eigen valu es of A? Is A di agonali zab le? If p oss ible, fin d an or thogon al matr ix P and a diagonal matr ix D so th at A = P DP ? 1 . A has eigenvalues 1, -1, and 0. A is diagonalizable because all symmetric matrices are. Since eigenvectors for different eigenvlaues of a symmetric matrix are orthogonal, we know the eigenvectors for eigenvalue 0 are orthogonal to both [ 1 1 1]^T and [1 -1 0]^T. Using for example, gram-schmidt or the cross product the only such vectors are multiples of [1 1 -2]. Normalizing we get P = [1/sqrt3 1/sqrt2 1/sqrt6; 1/sqrt3 -1/sqrt2 1/sqrt6; 1/sqrt3 0 -2/sqrt6] and D = [1 0 0; 0 -1 0; 0 0 0]. 2. (25) Let Q b e an ort hogon al matr ix. a) Wh at is Q T Q? b) Sh ow that det( Q) = ±1. (Hin t: What is det( Q T )? ) c) If x = 0, wh at can you sa y ab ou t || Qx || /|| x|| ? d) Sh ow that all real eigen values of Q must b e eith er 1 or -1. a) Identity b) det(Q^T)=det(Q) so 1 = det(I) = det(Q)det(Q^T) = det(Q)^2 so det(Q) = +-1. c) it is 1. d) If Qx = ax, then ||x|| = ||Qx|| = ||ax|| = |a| ||x|| so |a| = 1, so a = +-1. 3,4,5 See previous exam. 6. (25) Fin d an orth ogonal co ord inat e chan ge matri x P whi ch tran sforms th e quad rati c for m y 2 ? 2Ă2xy into on e with no cross pro duct te rms. The associated matrix is [0 -sqrt2; -sqrt2 1] which has eigenvalues 2 and -1. The eigenvectors are [1; -sqrt2] and [sqrt2; 1]. Normalizing and making these the columns of P we get P = [1/sqrt3 sqrt(2/3); -sqrt(2/3) 1/sqrt3]. 7. not covered 8. (10) Gi ve an exp licit exampl e of the singul ar valu e decom p osition of a 3 2 matrix with rank 1. (You nee d not mul tipl y it out). For example USV^T with U = 3x3 identity, V= 2x2 identity and S = [1 0; 0 0; 0 0]. 9. (60) Tru e (alw ays tr ue), False (alw ays false ), Ma yb e (some ti mes tr ue an d som etimes fal se, dep end in g on A, S , etc.) or short an swer. a) If th e characte ristic p olyn omial of A is ?(? ? 3) 3 (? ? 10) then A is diagon alizable. b) Two eigen vec tor s of a symme tr ic matrix are if th ey cor res p ond to diffe rent eige nval ues . c) If A is invertib le, then A is row equi val ent to th e id entit y matr ix. d) If A is similar to B and A is sin gular , then B is sin gular . e) If L: R 9 ? R9 is a lin ear tran sformation an d the kern el of L is {0}, then th e range of L is al l of R 9 . f ) If x and y are two vec tors with th e same length in an in ner pro du ct spac e, then x ? y is or thogon al to x + y . g) An y squar e mat rix A can b e wri tte n as A = S J S ? 1 where J is in Jor dan canon ical for m. h) For A, B , and C matrices , if AB = AC th en B = C . i) Th e eige nval ues of a symme tric matr ix are al l real . j) det( B A) = det( AB ). k) If A is an n ? n matrix , th en A is diagon alizable if R n ha s a bas is of eigen vectors of A. l) If A has eige nvalu es 1,2 and 3 th en A + 3I has eige nvalu es . m) C ol (A)? = . n) If A and B are non sin gul ar then (AB )? 1 = A?1 B ?1 . o) If A is an m ? n matrix with ran k n, th en th e matr ix of or thogon al pro jec ti on to C ol (A) is . p) If n vec tor s span an m di mension al spac e, then i) n ł m ii) n ˛ m iii ) n = m iv) You cou ld some ti mes have n ł m and som etime s have n < m, dep endin g on the vec tor s. q) If there ar e n lin early in dep end en t vectors in an m di mension al spac e, then i) n ł m ii) n ˛ m iii ) n = m iv) You cou ld some ti mes have n ł m and som etime s have n < m, dep endin g on the vec tor s. r) Ax á y = x á A T y . s) For an y matr ix A, th e mat rix A T A is symme tr ic. t) For an y matrix A, all eigen values of A T A are non negativ e. a) S b) orthogonal c) T d) T e) T f) T g) T h) S i) T j) T k) T l) 4, 5, 6 m) Null(A^T) n) S o) A(A^TA)^{-1}A^T p) i q) ii r) T s) T t) T ----------------------------------- Test 3 5/17/2004 1. (45) Let A b e the mat rix A = ? ?? 1 2 2 0 1 2 4 4 2 4 2 3 4 0 0 1 3 2 2 5 ? ?? a) Fi nd th e redu ced ec helon form of A. b) Fin d the rank of A. c) Fin d a basis fo r the colu mn sp ac e of A. d) Fin d a bas is for the Nu ll spac e of A. e) Fin d all soluti ons to Ax = [ 1 0 1 0 ] T . f ) Fin d all soluti ons to Ax = [ 2 2 1 0 ] T . a) [ 1 0 2 0 -3; 0 1 0 0 2; 0 0 0 1 1; 0 0 0 0 0] b) 3 c) [1 2 2 1}^T, [2 4 3 3]^T, [0 2 0 2]^T d) [-2 0 1 0 0]^T, [3 -2 0 -1 1]^T e) [-1 1 0 -1 0]^T + a [-2 0 1 0 0]^T + b [3 -2 0 -1 1]^T f) There are no solutions. 2. (30) For eac h of th e follo wing matrices : + Fi nd its eige nval ues and an eigen vec tor for eac h eigen value. + If p oss ib le, fin d a (p ossibly complex) matrix P and a diagon al matr ix D so th at the giv en matrix equ als P DP ? 1 . If p os sibl e, P shou ld b e orth ogonal . + If p os sib le, find a real matrix Q so th at the gi ven matri x is QC Q? 1 where C is of the for m C = a ?b b a . + Fi nd a for mula for 1 2 2 4 k as a pr o duct of at most 3 matrices . a) ? ? 5 8 0 0 5 1 0 0 4 ? ? b) ?4 5 ?5 4 c) 1 2 2 4 a) eigenvalues 5 with eigenvector [1 0 0]^T and 4 with eigenvector [8 -1 1]^T No such P and D exists because the 5 eigenspace is not dimension 2. Q does not exist since it requires non real eigenvalues b) eigenvalues +-3i with eigenvectors [5; 4+3i] (for 3i) and [5; 4-3i] (for -3i). So let P = [5 5; 4+3i 4-3i] and D = [3i 0; 0 -3i] We may choose Q = [ 5 0;4 3]. c) eigenvalues 5 with eigenvector [1;2] and 0 with eigenvalue [-2; 1]. Let P = [1 -2; 2 1] and D = [5 0; 0 0]. No Q is possible. 3. (15) A matri x A ha s singul ar value dec omp os iti on A = ? ?? 1/2 1/2 1/2 1/2 1/2 ?1/2 1/2 ?1/2 1/2 ?1/2 ?1/2 1/2 1/2 1/2 ?1/2 ?1/2 ? ?? ? ?? 2 0 0 0 1 0 0 0 0 0 0 0 ? ?? ?? 2/3 2/3 1/3 1/3 ?2/3 2/3 2/3 ?1/3 ?2/3 ? ? T a) Fi nd an or thon ormal basis for the colu mn spac e of A. b) Fin d an orth onor mal bas is for th e Nu ll spac e of A. a) [1/2 1/2 1/2 1/2]^T, [1/2 -1/2 -1/2 1/2]^T b) [1/3 2/3 -2/3]^T 4. (25) Find an orth ogonal bas is for Span {1, t, t 2 } in C [0 , 2] usin g the in ner pr o du ct f , g = 2 0 f (t)g (t) dt . Supp ose f (t) is a fun ctio n in C [0 , 2]. Fi nd the pro jec ti on of f to Sp an {1, t, t 2 } if 1, f = 4, t, f = 32 /5, t2 , f = 32/3, and t3 , f = 128 /7. Using G-S an orthogonal basis is u1 = 1, u2 = t - /<1,1> 1 = t - 1, u3 = t^2 - /<1,1> 1 - / (t-1) = t^2 - 4/3 - (4/3)/(2/3) (t-1) = t^2 -2 t +2/3. Then = = 4, = = - = 32/5 - 4 = 12/5, = - 2 +2/3 = 32/3 - 64/5 +2/3(4) = 8/15. The projection of f is then /u1 + /u2 + /u3 = 4/2 + (12/5)/(2/3) (t-1) + (8/15)/(8/45) (t^2 -2 t +2/3) = 2 + (18/5)(t-1) +(3t^2-6t+2) = 3t^2 -(12/5)t +2/5. (There may be an arithmetic error or two above.) 5. (25) Determine whether eac h of th e follo wing subse ts of R 5 are subspaces an d find a basis and dime nsion if they are. a) S1 is the se t of [x1 , x2 , x3 , x4 , x5 ] T so th at x 1 + x2 + x3 + x4 + x5 = 1. b) S2 is th e set of [x1 , x2 , x3 , x4 , x5 ] T so that x 1 + x2 + x3 + x4 + x5 = 0 and x1 + x2 + x3 = x4 + x5 . c) S3 is Span {[1 , 2, 3, 4, 5] T , [1 , 1, 1, 1, 1] T , [0 , 1, 2, 3, 4] T }. a) Not a subspace (0 not in S1). b) Yes, this is the null space of [1 1 1 1 1; 1 1 1 -1 -1] a basis is thus [-1 1 0 0 0]^T, [-1 0 1 0 0]^T, [0 0 0 -1 1]^T c) Yes, a basis is the first two vectors [1 2 3 4 5]^T, [1 1 1 1 1]^T. (Note that the third vector is the difference of the first two.) 6. (60) Tru e (alw ays tr ue), False (alw ays false ), Ma yb e (some ti mes tr ue an d som etimes fal se, dep endi ng on A, S , etc.) or shor t an swer. A and B are 8 ? 8 matr ic es, C is a 4 ? 8 matrix , an d S is a four dimensional subspace of a se ven dimensional real vec tor spac e V with an inn er pro duct. a) The eigen valu es of a Hermiti an matrix are all real. b) Two eigen vec tor s of a symme tr ic matrix are ort hogona l if th ey cor res p ond to diffe rent eige nval ues . c) Us ing th e usual He rmitian inn er pro duct in C 3 th e length of [1 + i, 2 ? i, 3] T is (1 + i) 2 + (2 ? i)2 + 9. d) If the characteristic p olyn omial of A has a rep eated ro ot, then A is not diagon alizable. e) If C has ran k 3 th en the null space of C has dimension 1. f ) There is a set of 6 li nearly in dep enden t vectors in P4 . g) If [u1 , . . . , u7 ] is an orth onormal basis for V , then 2u1 + 3u2 ? u4 + u6 , u1 ? 2u2 + u4 + u7 = ?5 h) V ha s an orth onor mal bas is. i) An y orthog onal set in V is linearl y ind ep enden t. j) (S ? )? = . k) (Nul A)? = . l) (AB ) T = . a) T b) T c) F d) S e) F f) F g) T h) T i) F (must all be nonzero) j) S k) Col(A^T) l) B^T A^T ------------------------- 5/18/2004 (partial) 6. (60) Tru e (alw ays tr ue), False (alw ays false ), Ma yb e (some ti mes tr ue an d som etimes fal se, dep endin g on A, S , etc .) or short answ er. A and B are real 8 ? 8 matr ic es, C is a com plex 4 ? 8 matrix , an d S is a fou r dimensiona l subspace of a sev en dimensional real vec tor space V with an inn er pro duct. a) Tw o eige nvectors of a symm etri c matr ix ar e orth ogonal if they corr esp on d to di fferen t eige nval ues . b) Us in g th e usual He rmitian inn er pr o du ct in C 3 th e length of [1 + i, 2 ? i, 3] T is (1 + i) 2 + (2 ? i)2 + 9. c) If the char ac teristic p oly nomial of A has a rep eated ro ot, then A is not diagon alizable. d) C C ? is diagon alizable. e) There ar e 6 vectors in P4 whic h span P4 . f ) If {u1 , . . . , u7 } is an orth onormal bas is of V , then {u1 + 2u2 ? 5u3 + u4 , 2u1 ? u2 , u1 + 2u2 + u3 } is an orth ogonal se t. g) If A has no real eige nvalu es, th en A is nonsin gular . h) V ha s an orth onor mal bas is. i) An y orthog onal set in V is linearl y ind ep enden t. j) dim S ? = . k) If C has ran k 3 th en dim N ul (C ) = . l) If A and B are invertib le, then (AB )? 1 = . a) T b) F c) S d) T e) T f) T g) T h) T i) F j) 3 k) 5 l) B^{-1} A^{-1}