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Math 205A Exam 2, page 1 October 7, 2005 INITIALS. sss 1. Suppose T and 8 are linear transformations and A and B are their corresponding matrices, where A and B are given below. 1 5 -7 S 12 3 A=| 4 21 -24/B= 3 2 : 2 —+s |e —> roe bon) BR R°— IR la. S: R* + R”™ where k and m have what values? m2. 1b. Explain why you can’t form the composition, To S. The cota" fot vechrs ty MR* gk the etormind Ti 3 le. Let ej denote the i-j"" entry of the matrix of the composition ST. Explicitly find c23. Cr 6 obfahedt by mollpilying roxs 248 by Coban 3h A! [ ane I: a (2) a Pt G24) #1229 = 21-1944 204 v [15-4] =-2 -? 1d. What if any conditions are there on by = i so that b is in the image (or range, as the 2 book would say) of 3, ie so that S(v) = b for some v in R*? Show all your work. This eyaalnt oon [SE AIR] ~[5 3 3 Lae 3642 O00 3 hm dew B= ~ [oot] 52] hive & sol 7” As relechon yields z . 4 ve very b/ | a ee le. Is Sa 1 to 1 linear transformation? Explain fully. bf 4 4 Lg fone o- are ne incoaustencite bon Mo, Si)=8 amt 2 MP. Gi) 8 tng sobs bE sci he mobic fr Bia Ru=d has cont) Soles bf (ont cedure y ickls 4 free cial, ((e%/3)~[2 4] 2] Shows xyes ag . 1f. Is $ onto R™? Explain fully. any veer o} the feo xf] GMs YES: in Id we Sew Hat 4 ie NS a Beg Bere ele image 4S, 2 3 thet oy fC onto R™ , (ioe Herod eed sr IP fn Lent bb bs se Sa Mor 7-/,) Math 2054 Exam 2, page 3 October 7, 2005 INITIALS da, Give an example of a matrix in My» which has four different, nonzero integer entries and which is not invertible. we reed the dhlermiunt & be O, 2. +t toy fa 3 a br examples 4b. Give an example of a matrix in My» which has four different, nonzero integer entries such that the inverse also has four different, nonzero integer entries. yx rook he chernitook to be L for extemelif A= 32) te A ges Sek - [23] 5 What is the definition of Linear Independence? “A set S = {v,,Vo9,...,Vp} is linearly inde- pendent if and only if...” al the goby Liner conte fee : AU take... + Oe oid pele © bo jee ane intduch alll veghtt % Oo 9 "9 Mo = © Math 2054 Exam 2, page 4 October 7, 2005 INITIALS 6 Consider the system of equations: a +4r 4232534 2024 = —4 32; 4+ 139. +7323 + 6524= —-19 5a, ~ 17%.—102z3 — 88ry = 15 aj+a, +15¢3—-Try =81 Use the information on the front of the exam to answer these questions. 6a. Find the row reduced echelon form of the augmented matrix corresponding to this system. [ 1 oo al-se o lo o th] cH oot -3 16 C 0 60 Of @ 6b. Write the complete set of solutions in the form v, +p as done in class, that is, where v;, represents all solutions of the homogeneous system Ax = 0 and p is a particular solution to Ax = b. , x (- a it have. \ 2 He] fe xy {lB + [PL] alee xy 6 thee. Xy L! ° Gc. If the 81 is changed just a little, to 80, what is the RREF of the resulting system? (you might have to do a little additional row reduction and you might not) 2 | -8& Loam he inf, fost come: "¢ I O11 of|-2 hid. in PREF io ‘Save 3 Lai coUs|4 ° an Ebdon brme | Sno ol | 2 eee | 6d. What is the complete set of solutions for this new Spe again in the form v,_ +p? nigw the BREF reveals fan ircersislany tn bast (rt 5 thine can be AO SOLLTION 6e. Do the columns of the coefficient matrix A span R*? Explain. DO. ter example, Gol hos Mat “i hae « LE f &O He cobmns J A, fo Heese cobinne ely wot Gag Led R" 6f. Are they linearly independent? Explain. fo. the he yanielle Xy TLE tio Had — Ax=6 bec 09-me a me fo aca tne fo gti T be > 
