Question

Mark each statement as True or False and justify your answer. a) The columns of a matrix A are linearly independent, if the equation Ax = 0 has the trivial solution. b) If vi, i = 1, ...,5, are in RS and V3 = 0, then {V1, V2, V3, V4, Vs} is linearly dependent. c) If vi, i = 1, 2, 3, are in R3, and if v3 is not a linear combination of vi and v2, then {V1, V2, V3} is linearly independent. d) If u and v are linearly independent, and if w is in span{u, v}, then the set of vectors {u, v, w} is linearly dependent. e) The columns of any 4 x 5 matrix are linearly dependent. f) If S is a linearly dependent set, then each vector in the set is a linear combination of the other vectors in S. g) Two vectors are linearly dependent if and only if they lie on a line through the origin.

Answer 1

PICTURE 1 : - A, B

TRUE A set of vectors Evi, V2, ., Vn} is Incandy independent if @,V, +C2V, 4 • *-+ Con Vn ) Crl2...Cm are all Lero. Now suppore A be a matrin whore columns consist of the vectors V, V2, ...,Vn Then note that C, Vitex Yz . + Cm Vn so can be willen cis (6):68). So having a trivial solution d (2) is equivalent to linear independence of vectors vi, v₂, 2... V. as in (1) ie the lineær indeféndlince of the columes of the matrise A. 3) TRUE Consider the set of vectors {v,,V, V3, V, Vs} in its with V3 =0 there equirt {vi, V2, , Un, Vs} cvill be linearly defendent if sealary 6,,C2,C3, C4, C5 NOT ALL ZERO such that e, Vrt GV2+€3v3+ Con Vnt GV5-0 - 00 . Now since riz in the zero vector & any sealar times the zero vector is 2000. let C, ,C2,C3, C4, Co be zero. This does not contradict (i). but het C₂ = 1 Then also the constants , C2, C₃, C4,6 cont contradict (1) So {v. N. VA CO), VW, Ys;} is linearly clepen olent.

PICTURE 2 : - C, D

c) FALSE Vi, i = 1, 2, 3 EIRS V3 is not a linear combination of vi & V2 Conycler v;= (1,0,0), V2=(2,0,0), V3 =(0,0,1) clearly 3 is not a combination of V. & V2 Now evit.cz Vz + C₂ V3 20 >> C,C1,00) +6 (2,0,0) + (30,0,1)=6,0,0) *)(,+227, 9, 9 ) = (0,0,0) :-) @ +262 20 = 6,²-22 C3 26 V,V,V3 are not knecerly independent. (Note that ; Virve are linearly dependent). d) TRUE Verare lnearly independent. Wespari{u, v} Then w=eutev (a , cr are.). - scalary Connicker the equation. d, ut da u + dz czo (didzidz are scalars) on diu+dzvedz (C, N + c2V) =0 or u(d,+ d₂ a.)+ V (de + d302) =0 or ditze, zo Tour are linearly 7 . & dat dz C2 20 L so d, = -0.dz . . dz-zdz do , dz, da may not all be zero. 3. U, V, W are not linearly independent Independent !

PICTURE 3 :- E, F, G

TRUE Note that the column vectors of any matrix A in linearly independent of the equation An=0 has only the trivial solution. Hence the columns are dependent if VARZO has a non zero solution. A is a 4x5 matrin - It is equivalent to a homogeneous system cohich has 4 equations & 5 variables this syitom has nonzero solutions (infinitely many) So the colemms of A are linearly dependent. FALSE Consider the enamble in part e) V=(1,0,0) : "2=(2,0,0) : N₂=C0o,1) We saw that {v, V2, V3} is lnearly dependent but V₂ in nota linear combination of vi & V₂ because ani +22%2 = 72 = 9+222 200 do ai imporrible!! 9) TRUE If the two vechoss be on the same line through origin Then the origin in the combination of Here levo vector. . thus they are linearly dependent. The combination to non trivial 1) conversely if the two vectors are linearly dependent then they are medtiple of each other. Hena they be on the same line (v, ev live on the line y=ex containing the ne "brigin !).