Test 1 Math 2407 Spring 2011 1. (9 points) For each of the following augmented matrices answer the following questions. ~ 1 0 0 5 6] i. [ 0 000 1 X( 1('1 7ty: Number of equations of the corresponding system is 'J. i Number of variables in the corresponding system is _... '7:<--_ True o~: The corresponding system of linear equations is consistent. b. fp cd ~1 ~ ~l l~ 0 0 'JL 1. )t\ 1-, 'x,,; 1-r Number of equations of the corresponding system is _~_ Number of variables in the corresponding system is _...z:::.:...-..._ Qor False: The corresponding system of linear equations is consistent. c. [H ~j 000 ;L ")(l- Number of equations of the corresponding system is!y( Number of variables in the corresponding system is.2.. (3e or False: The corresponding system of linear equations is consistent. Tl--tAy
2Xl + 2xz +2X3 = - 2X3 2. (12 points) For parts a-d consider the following system of linear equations -3Xl + Xz -1 = 0 2.t1 +;L{:;1 T 4. 'J?> co SXz + 3X3 = -1 -;;:t l -+ 7:."'- = ~- 7(;; + -3 X::; =:-j i. Write the coefficient matrix corresponding to the above linear system 2 r; Write the matrix equation corresponding to the above linear system ~]... [~~. J ; [ ;] [~ r? 't0~ - i Write the augmented matrix of the above linear system [-: 2 f. ~J 0 S- 3 iv. Row reduce the augmented matrix of the above linear system until it is in row reduced echelon for TL, and indicate whether or not the system of linear equations is consisten~r not. () l o o 2..L,;( - 'L 1(>! o o -) f.-, ~ o () R;;)-~l-'$ -y~~ o D t rwjl' r.ecl:ale i 2.. L ~efe", f~ ('$
3. (9 points) For each the fojlowing~ describe the solution set in the indicated forms, of the system of linear equations corresponding to the.augmented matrix. [ 1 2 0] i. 000 :L{ Describe the SOlutiotSt by describing the free variables and then the non-free variables in terms of the free vari les. Xl:::; -.:1 7-'1- X2 = i ~ fvle.. Write the solution set as an ordered pair. l_'-_:l_x_~...-.:.)_7_c...=-:)--'-)_'j r l. 1-'J. -;{~ [-]2.1 Write the solution set as a vector equation _x = -l_=--..1...+~.l_-'-_j.l:-:;:.l..l_ X':: (<' J 'C, [~2.]C',,,,?""l- "'" "l-2[-2l J --,,"v J! Describe the solution set by describing the free variables and then the non-free variables in terms of the free variables. Xl :::; P --\- l-1-).. X2 = S 1-y.t.e; Describe the solution set as an ordered triple. _-,,-3_+_:L_1-_1---.-:: ~S_))..; 1-),. Describe the solution set as a vector equation X=[7<.1 'l.;- J C('+.J.X"J il -:;0" [3] +r -,{2..1 ''> 5' S- 0 4. (5 points) Give 9J:l.>2@~ oj~ql!~ns in 5 variables whose general solution is (u, W, X, y, z) =(t-2l- y, W, x, 2 - X, 3 - X + w) [~ ] ~ rht~ lj < ~ ~ _,:.:--a ~ L;~~fCV i: -:: ~ -)CT_vJ 'vv VA d 'X- V"~. -f ~,. (,{ -2. ll +~ ;::.. \ - ~-+ ~ ~ ~ '1., - vj ti. ~} j
5. (3 points) Let A be the matrix whose columns are [~~:~] [~~:~] and [~::~]. Use the definition of multiplication au az,3 a3,3 of a matrix by~to write out A [~:]::;: [~:~ ~:,~ :::~ J' [~~J... 3 A,% h., ~ u."t'; 1.,. [ 6 2 1] [-3] [-13] = c, = c, = +3 6. (3 points) Note that '-:1 ~ ~ i = 5. Use this fact and no row operations to find Cl f C2 and C3 such thaf:} c{!} c, [~] + C3 [H C, 7. (6 points) Write each olthe following vectors as a linear combination of the vectors ([~]. [~l [~ll i. [~l= >[~ ] + 0 [~]+ [U. 0 i [~l= 0
8. (4 points) i. Give ~~r in R2 that is not in the span of {[~], [=~]}. f no such example exists, explain why that is the cas.l. A., ~~~l~: [~ ] ;y [--] + i~ [=~1~ [St ] ~.-rt [L ~ 31J~ [: - :J 0 }1[f2-J+1~f:t] ~[~] 't' ': ~ Q fa:ffola ~f.., \ (3] 15 k(). q.. l.,:f<.>o-r Cc>""'~rV\~~\bn of [~JO-.J [~]. ~~, [1,] S ~04- '~ 4e!r-~ of ([~J, [~]} i. (4 points) Give an example of a vector in R2 that is not in the span of [[~], [~]llf no such example exists, explain why tha. t is the case. '" l- d _L ".L L ~OO- f~ liu)ea~w-:pr<l!!'t,. 01 b... V, Ct;O {r' ;,.,. (1(... ~~ is 1f6'f {'" """" L 1 {fr'y\ "f t L! J GJ 1. L~ ' h!:>-[~j ih tt' t-.'" J 9. (8 points) Let u [!] and v = [~]. Graph and label each of u, v,iu, -u, u + v and u v in the plane. ;j:u :: t [~J [~] C'J -~~-[~J~[~J ~O <A+V= [~J+ C:J ~~J
!A "-- 0\;.J J..v 10. (12 points) Determ ine ilthe vectors [ ~41'[f;] and [JJ spa n R'. (you m ust s how the work and e xp la in answer) [ +; [ -; f [l 1 MAtY'(X (?] -i: -l t"0j ke [t'l c\.. ) l!t r2-1 1 a o $'" J""\ 0 ~ ; U] V [~ 0 $"" '-P-l -1?--<. t[tl " J j, 0-8 -t\~ ~i h~ 0 0 -til.} -:?\4-W~ --'j«'j ~'l [ttl t\').. LA? J ho-> (;\ ptvot. po'sr-e(ovl (t-\ J2.\ 'ry Vow) So ~. 'iz't,!),ct,tibn (fa "'2- CAl J y: :ch ktt.$ fa Jo u:.;uo V lkf~fot'q) [~i 1, (~J ~J [~] <;" rlh..y\ ". n J. l 11.(15 points) For each matrix below, determine whether its columns form a linearly independent set. Give reasons for your answers and make as few calculations as possible. (0 (vtwl.' lvld Me i t.lt <lv' <> f ~..,... 15 "' We... ~G'-,e, tjo V1OV1~lro, -4 12J ~e{l'..,.f.()~) *h~ two 'eol(.(""'h.5 &,-.V'(! [-3 8 'J.::: -~)<'r t) t~~:~ =:r~~' a. 1-3 WtA (tl-p'e af tf,te;' o.4.cj-y. '~'" -1; ' ( ) "6 i -l, '.,(.'l,) [-:4 7 ~j 1 i\~ ~ r\dlth')( (ol'\'te<i t'\ 'St;tll\ t;'avc lo(..."" Vi, ~o b. 6 Co kt."", V\ S <l\.\r ~~PY!J d~~~~. 13 1"- r o.t ""1- ( ) 5 3 O-Vld 4. ltv..., h'w' '01 ~lct"e.. (SJ t ) VA".s c l~ 4 9 0 0 1:J (", ~ e!ll&~'~...f ~J"...,. ~Ve.. ftljn'v,<;;. 3.tJ(v«~ 412.- VV1tt-bf')'- ~C.~ ""V-~~~ k~ -tk~ WY Lolu...,,,,s g-ye. ~~ (f,.'re,. e~f::"res of (2,~c..~ col"v1y1h, ~-~'-~--~-- 1
12. (10 points) Circle your a~wer true or false i. True or~jany system of n linear equations in n variables has at most n solutions. Tr~e or~; ",sider a system of n linear equations n m variables. Then if n < m, then the system S fnconslste~vl i True or GjA iso 1 x 4 matrix, then Amay have 4 pivots. iv. True or~lhe augmented matrix [1 0 2 0 -lj corresponds to an inconsistent system of linear equations. J', 7<~ :',-;rtf 3 VO<i"i",bie5 v. ~ orjalse: The homogeneous equation Ax=O has a nontrivial solution if and only if the equation has at1east one free variable. vi. True or ~ f x is a nontrivial solution of Ax::::0, then every entry in x is nonzero. v vi GAr False: f A is a 3 x 4 matrix and x is in R 4, then Ax is in R3. ~ fd p- 91 False: The solutionset ofax=o is R3 if A is the3 x 3 zero matrix. ix. ~r False: The zero vector in the span of any collection of vectors. x. ejr False: f u and v are in R m, then -u is in Span{u, v}. (0