Problem 1 9 Solve the linear system x xz 5x3 1 2x 242 943 4 x mx2 34z 4 Solution Write first the system in form of A 5 A 3x3 tIR3 A 5 2 l a R R 5 R3 R2 R2 2Rs I O I Je oI O x 14 22 3 xz 2 1 b Is it possible numbers I 4 4 on the linear system that the Explain system becomes briefly to change right in hand side of such a way inconsistent No as A has pivot in every row 1 c Give a non trivial A o or explain solution of why none exist No are non trivial free no variables solution as there I d True or False The columns of A are linearly True independent I e True or False Span R3 True J The columns of A I f True or False is always augmented matrix in pivot position False The system A 5 consistent if the A has a everyrow Problem 2 Given A matrix M below the RREF form R of 2 M 3 O 0 30 8 IO o I o 130 o D 2 0 O br br fr bu gj r 4X5 o 0 2 a Consider the linear augmented matrix is M solutions this of form rector system whose Describe all system in parametric x 6 44 xz 8 343 u3 free xy 2 2 um particular solution of AX 5 general solution of Ax of 2 b the Let A be system whose Write down the augmented matrix coefficient matrix of is M then describe all A the and homogeneous solutions of in parametric system A vector form and in spant 3 A 3 51 3 21 I 3 110 1 6 I 0 13 1I Y xz 4 Y span3 True or False The columns of M are 2 lin early False independent FalseThe columns of M 2 d True or RRY Span False or False The columns of A z 2 e True span IRY False 2 f Is the linear transformation T R5 R is one to one Explain briefly No every column as A does not have pivot in 2 g Is the linear transformation T R MY is onto Explain briefly No in every row as A does not have pivot Problem 3 3 a Let T R3 R3 be the linear transformation defined by T Find the standard N matrixe ofT A T e e e ei b ii T e T j es j A g 8 3 b Let Ti transformation R2 IR3 be linear a which satisfies 0 2 i S ii Write Standard matrix A of T A sx ii Evaluate T 2 2 2 i s T asT x Ax ii a Find T x 5 Solve A 5 rector such that where 5 4 for b s x 1 u 2 3 2 Find the for standard matrixe the transformation S R2 R2 first reflects rectors through and then stretches Uz x scalar factor of 3 linear which the line by A T eY 2 5 n Tr 12 R2 TaIT As A Ts R 2 IR2 Ts x B B 185 T I R2 T ToTr x Ts Tr Ts A B A BAY Problem 4 a Consider the linear system with augmented matrix n 3 values of h for which all has Determine the system i exactly ii No ii Infinitely many solutions one solution solutions m 0 jin if No solution R2 R2 hR 4 h2 0 and n 2 4 and n 2 and 6 3h 0 3h 6 n 2 ht 2 M 2 b Exactly n n2 0 one solution net2 a Infinite many 4 h2 0 n 4 solutions and 6 3h 0 and n 2 n 12 and A 2 n 2 4 b Determine if linearly the set of rectors independent or dependent Justify your answer are below linearly 9 2 i l iii do6 of 60 O O Linearly dependent ii 57 7 ii linearly dependent iii 8 18 8 linearly dependent iv 1 10O C Linearly independent Problem 5 Let A 3I 0 L 000 5 0 4x3 0 5 L 4 I 9 Give the number of solutions of linear A 0 system I solution G b Give the number of solutions of linear system AX b 0 solutions 2 Letai an above and as denotes let5 be the given above i True or False of vectors is The linearly True 12 3 I I 0452 006 3 00 o 4 eme Ox 0xz 0xz Y columns of A the given rector set Sai 3 independent ii True or False of rectors is The linearly True set Sai 5 independent 2 iii True or False The set S 3 of vectors Span R 1 False iv True or False The set S 3 3 of rectors True Span RY A oO S Problem 6 b Problem Consider the vectors ii 3 i 32 107 span uP span a 2 net has a solution for i No b 2 dependent through linearly i a line ii a line iii iv a a plane plane through through through in 13 origin in M3 origin origin in As 13 origin in A 2 x 3 linear transformation T R3 R2 is onto Problem 8 a True or False Every False b True of False an uxu is independent system If A linearly linear the is consistent True for every matrixe with columns then Ay 5 in RM c True or False columns span If the A transformation defined of an nxn matrix then the T Rh linear by T AT is one to one True D d True of Fake SF T R R2 origin through an transformation TRUE rotates rectors about angle O then Tis a linear Problem 9 Let A S 75 2 1 for find the value Such that linear is consistent and the parameter system Ax 5 give the solution t 2 I12 xz 4 xz 0 t 2 O x I if 4t 3 0 Consistent only i Problem 10 Let S 1 transformation 2 be the linear defined by Anxzs mi x x 3 4 4 53 4 x 3 4 22 a Find the standard matrix of S Standard matrin of s s e S e T in m Amxn S d i 0 s i standard matrin in C 5 b Determine whether S is one to one Justify your R2 R2 53R No answer 53 4 O N O Problem 11 Let R2 R2 transformation You are given 2 2 be a linear and i 5 Find linear the Standard matrix for the transformation Solution mint d i 2 2 i 2 2 i I 2 2 i c 2 j a b 2 standard matrin T1 0 i 35 Problem 12 Solution Reflection transformationinisin matrin id J Matrin for Rotation applying reflection transformation transformation after sinDu cos to I COSTDG SiriyG cos 1T2 1Mo sin T Mc I sins I O 13 3 R2 2 5 3 53R Alternate Method Rf Ro R2 R …
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