Question

0 (1 point) Let uz = 4. → U12 = ,113 = 4 Which of the following are in the span of u1, 12, uz? 6 A. 8 4 :-2 B. 8 1 C. Write the following vector as a linear combination of u1, U2, U3. If it is not possible, leave each entry empty. + () + [:1-03 3

Answer 1

Let u, O 43 0 -4 1 - lue have to determine, which of the uv, U2143 ? follocuing are in the span of A 6 [ . B. 4 We vectom x be يا أبت 2 Vectors wie was up, U₂ 43 one That is / Know that, in the span of a W = set given of a lincan if combination of the vectors w can be written w = 9,4, +C24, +63 43]; if w is in the span of 4,421 0 U₂ and 41, (2.62 are constants. words say that w = 6, 41+ C2 U₂ C₃ 43 be in the span of u, ,U₂, U3 if can find at least one of solutions for the coefficients 4,4ily In other eue can يا أيه set Fon the given problem, we have to check for theree Vectors (A) 6 Here we have to check if can be in the linear span of or not. Let us write; for constants 4,4 and 3 ; 8 4 4 0 +3/-4 o 8 G + C2 aa-463 C + ₃ 4 G+C₂ = 6 -(1) 44-463 = 3 and (+ €₂=4 G-6352) -Liig Subhracting (i) from (i) we have which 9,-63 = 2 is identical to (wi)

So we have two equations G + C = 6 and q + C₂ = 4 the system is redundont). For three unknowns e, and lz have tovo equa Hons so Choose one variable arbiharily. ( as Luc eue can Let us Chouse Cy=0 then from q + C₂ = 4 have 24 And from , G+C = 6 lue have 4 + 4 = 6 on 9 = 2 Thus Lue have a solution set for 4, C i Cg ie, c=2 ; C2 = 4 and 3:0, Hence 6 is in the span of ui, uz Uz. (B) In this case w [:] . we have to check if w can be written in the form of (i) -4 From ); 010-11-19 a 2 4+C2 401 - 403 C₂ + C3 a, C+ 2 = -2 44-413 = 8 ; C2 +63 --4 9-42-) (ui) Subhaching (uü) from ; we have; G-13 = 2 which is identical to ui). wu So, have two equations 4+2 = -2 and (2 + ₂ = - 4 for three (the system is redundant) So, we can one variable anbi barily. unknowns 4,2 C₃ Choose

let us chouse, then from C3=0 Cztez=-4 OY (2 = - 4 from, a+₂ = -2 oy, cy-= -2 G= 2 Thus, cue have ܘ sow Hon ser for C1, C2, C3 such that i C2= -4 G=2 and C=0 Thus, 8 is in the span of u,, U₂, U3 (c) Here we have A] to check if w can be written in the form o ie, if w is in the linear span of c,, U₂,43 or not From ; CU O 4 0 4 % a +43 - [-] C+22 46-463 C2 + (3 . 5 49-493 = 1 (2+(3 = -4 (6 + 6 = 1 - fül) 9-C3 = V4 fix) Subhacting ) from fuüi) i we have . (xi) ( = 5 From (2x) S; that 41-63 = 1/4 , the system of equations is inconsistent because there is no way that 6,-43 com equal to 14 and 5 at the same Hime. So, we can not get a solution set for 4,62 € Such that w=auI+quq + Cq Uz for w= = [!]

Thus, [] is not In the span of 4, Uq:43 So, The vectos which are in the span of Up , M , Uz are; 6 - 2 A . [] and B. 8 Νω 6 Den be cwritten as ) [4] -ta[:]--[:)--[:] [sion ne marcos]