2nd Midterm exam MATH 2331-Linear Algebra Summer 1 2015 To obtain full credit, you must show all work and carefully justify all assertions. ‘The use of notes, books and a calculator is not permitted. NAME: (1) Consider the matrix 012030 7 (0 0 0 1 490 000001 ov0000 1) (5 pts.) Find a basis of the Kernel of A. Fam vorisbls x, xy Xe s 3 xa 0) ol | Se Lot | . ro bosis vectocs FO,xs2/, Rye % Vea for leer (4) ooo Ke 20 Eel (b) (6 pts.) Find a basis of the Image of A. “C veetos with leodieg vaortbles i 0 7 7 . (3] ie} Vee : ¢(2) (10 pts.) Determine whether the following vectors form a basis of R* 1 1 1 1 1 -1 2 ~2 wei] w=] w=ly) w=] 7 il -1 8 -8 ' ' 7: ' q \ ' a ff _— Z - a invert ble: {el 8 ee fi ion 1 | Bo] 2 ee | g ;2 9 3.0 m1 9 0 3 »o \|s3 ~ a. a terangular vith nonvero Miagonal votre 5 A is invertible Vn basis. (3) (10 pts.) Find the orthogonal projection of 9¢, onto the subspace W of R! spanned by the vectors 2 2 1 0 voy, = -4444 040-0 > o1¥ho gon ah. Whe (eT 3 = hm ye by and ue by, altered > pol [4e,)- (4(4) Consider the following vectors in R*: 1 0 1 walt ols le ae th 2 1 G 7, 6 (a) (Gpts.) Perform the Gram-Schmidt process on v,, U2, Us to obtain orthonormal Vectors 1; Up. iy v 4] ( tye Tags eg 1a to 0 iv: [1] \. is i !Q-0 & a (b) (4 pts.) Express your answer as the QR factorisation of a matrix V (note: you are simply asked to give the matrix V and matrices Q,R such that V = QR.) ni 0 4 : = . is Ue 4a 2 16 10 1 io Vie rif ware go |[o 2 ve Mo(5) Let 9,6 be two fixed angles and consider the matrix singcos@ cosdcos —sin# A=| sindsind cosgsin? cos? cosp | sing 0 (a) (5 pts.) Show that A is an orthogonal matrix. Oe a a sinh 69 sin bsin® cos Sing Od cord cont -sin Coo} 39 (orp Sin - Sing Sind So os gan cr in cos . et tind 5 (b) (5 pts.) Find the inverse A~? of A Sia & wos B En hsin® esd at. AT = (2 b Lot e4 Sin® — ~ Sin 4 ~Sin 8 or? 0 (checked vs pe ca)! )(6) Consider the system Ax = where i ‘a =(10 and b= [3 o1 3, (a) (7pts.) Find a least squares solution 2* of A, x*. (ATA)'ATL. 5 (b) (8 pts.) Is this solution unique? Justify your oye so is solution of Ax -ATb yes since ker (ATA) = oy > {0} syslens 5 consist, ond har auc gent & cosk 1 —sink (7) Consider the matrix A=[ 0 2 0 lution ! sink 0 cosk ‘a) @pts.) Compute the determinant of A. ak (A): Zork .O10- Cassk)-O-0 = 2 (b) (Spts.) Determine the values of & for which A is invertible. deb (A)22 40 Fo all k jm Ais sve tible for all values of k.(8) opts.) Compute the determinant, Laplace cxpantite 00300 j 00002 jd 300 420 04000 -+5it 328 Bicol |e cal goo1o Vb ojo gR Vo 5-0.0-0-0 ie : = 25 a). 543-(.2)2 -l20 (9) (10 pts.) Use row operations to compute the determinant ee ! -l -1 21 6f4r o | 20114 10}-2r Po = _ -2 6 10 33 }4eT \ Y io g ft) ae fii | |e 7 : | os 4 feo io hl o \ 2 Os cm | 0 ° ° 4 | =r dk A+ 4 (product of dingord extnes of han quter matrix)(10) ‘True or false? (No justification is required for your answers to the questions below. A correct answer is worth 1 point, no answer is worth 0 points, and an incorrect answer is worth -1 points) (a) Ifay,...,¥, and ay,...,t4, are two bases of R", then n must be equal to m. alt bares of R° wust hove \ the same Catdinality «10 (>) 1A is a 5 x 6 matrix of rank 4, then dim Ker A is equal to 1 tank nelly thes FE lin (ter I vein ie )= 6 ~> dim(hesP) = 2 cose T®D <4 (c) The image of a 3 x 4 matrix is a subspace of R! cabspare of RY (a) Ifyy,...,u, span R', then n must be equal to 4, by YW Twill oe. yyreker thea, of ered ty FE nell only sf Wa are | early indepen duct (e) There exists a 5 x 4 matrix whose image consists of all of R°. by rank nullity thm, 2 dion Cart) <4 (2) If the vectors 11, 9,¥,24 are linearly independent in R", then the vectors 4, Ug, Ug must be linearly independent. if CM Oh cry, -O, wha ¢ 7 Youn V+ r¥2t C3¥yt Ovy - 7 (g) If the vectors v;,u2, ug, U4 Span a subspace W of R", then the vectors v;, Us. Uy must be also span 1. re 9 ra |] (h) If a subspace W of RS contains the standard vectors €;,€0,€5, then W must be RY _ din W = cank(e,,¢, € 2 ray ce i ie Yume, 2, Span Sima Wee Subspace (i) If W is a subspace of R" of dimension m, then m < n. basis of Whas im elommts thot are indepprdak ve RO o> men (j) If the kernel of a matrix A consists of the zero vector only, then the columns of A must be linearly independent. : A: Cy > : C Cvt 4 Cay = since lee @ = 49], tha colomns must be linearly ince por duct and fe hua