##### Collection of Solutions [Contribution appreciated]
thaiarthur
I thought it would be helpful to get a thread of solutions going for convenience. Please try to keep discussion of specific questions in different threads, to avoid clutter. I am still trying to figure out a lot of these problems, so please do let me know if anything is wrong, and please add missing solutions if you have them! Missing: Image for Problem 5, Problems 6-8 Problem 1: 1 0 0 7 | -34 0 1 0 1 | -5 0 0 1 -4| 21 OR -34-7t -5-t 21+4t t Problem 2: RREF: 1 0 0 -7/6 0 1 0 17/3 0 0 1 -1/2 Dimension of image: 3 Dimension of Kernel: 1 Vectors that map to 0: 7/6 t -17/3 t 1/2t t Problem 3: A inverse is 2/3 -1/2 1/3 -8/3 2 -1/3 1 - 1/2 0 Problem 4: (a) Linear (b) Not Linear (c) Not Linear (d) Linear *UNSURE* Problem 5: Im(A) = Ker(A) = -5/4t 1/8t t Problem 9: [T] from gamma to gamma = 3 33/5 21/5 2.5 114/10 31/10 -1.5 -82/10 -16/10
thaiarthur: May 5, 2015, 2:18 a.m.
Formatting seems to have not transferred over. (If anyone can show me how to properly format this, I would appreciate it) Here is a link to a google doc instead: https://docs.google.com/document/d/16ct_xvWp2PaCJEV66W1ww-fxVpoRoVTDyKCr2up38eU/edit?usp=sharing
thaiarthur: May 5, 2015, 3:27 a.m.
Update: I have at least an answer to all of the problems. Please compare these solutions with your own and post any discrepancies! I would like to get a solution guide going for all of us. Specifically, if someone is confident on their problem 7, please lend a hand!
Jackhe: May 5, 2015, 9:13 a.m.
I don't know latex so I use this website to write equations in latex http://www.codecogs.com/latex/eqneditor.php
akaur: May 5, 2015, 11:40 a.m.
For # 7, I got 7/2 1/2 in first row and 2 0 in 2nd row.
akaur: May 5, 2015, 11:43 a.m.
Forgot to convert it to (a, b) so here goes: {(7/2)a + (1/2)b, 2a}
wllmskn: May 5, 2015, 1:08 p.m.
for 7 I got {(3/2)a + (5/2)b, 2a} but I'm not entirely sure either
thaiarthur: May 5, 2015, 2:39 p.m.
I got help from a TA and agree that Problem 7 should be {(7/2) a + (1/2)b, 2a}
yulduzkhonbruin: May 5, 2015, 8:18 p.m.
I got the same for problem 7. T(a,b)=(1/2(7a+b), 2a)
Anilysis: May 6, 2015, 12:01 a.m.
Problem 1: x=-34-7w,y=-5-w,z=21+4w,w=w. Problem 2: dim(Im(A))=3, dim(ker(A))=1, ker(A)=span(7/6,-17/3,1/2,1) Problem 3: A^-1=[(R1:2/3,-1/2,1/3),(R2:-8/3,2,-1/3),(R3:1,-1/2,0)] Problem 4:a)linear b)not linear c)not linear d)linear Problem 5: Im(T)=span((1,3,1),(2,-2,0)), ker(T)=span(-5/4,-1/8,1) Problem 6: B is a basis for P_2_(R) because the vectors span P_2_(R) and are linearly independent. Gamma is not a bases for R^4 because the vectors aren't linearly independent (the 4th vector is redundant) Problem 7: T(a,b)=(7a/2+b/2,2a) Problem 8: [T](Beta->Gamma)= [(R1:1,-2,0),(R2:1,1,1),(R3:2,4,0)] Problem 9: [T](Gamma->Gamma)= 1/10[(R1:30,66,42),(R2:25,104,38),(R3:-15,-82,-24)]