Question

Problem 1. Let V1, V2 be linearly independent vectors in R3. Using definition of linear independence to show that {3v1 +4v2, 4v1 + 7v2} is linearly independent.

Answer 1

Solution Page-L Given ارا ） are We have to prove a oure Νοω, linearly independent vectors. {3v, +4V2, 40, +7V2 } lineary independent leftis understand the definition of lineary independent vectors » If a set of n-non zero vector X, a to linearly independent set of vector, then If the linear combination of the vectors is Zero vector King+k₂ łą + K₃ Kg -tkn l=0 are then it is necesse essarily twe X* Ik, = K₂=kz= =kn Soi 1 we can write the linear combination x, (3V, +402) + x2 (4V,+7V3) = 0 > 30 V, + 40, , +48V, +72₂₂ =0 >> (34+492) V, +(4x + 713) V2 =0 It is ginen that Vi & V₂ are linearly independent 34, +422 =0=42, +742) So,

Page 2 solving both equations 4+(32, +422=0 3x144,+722=0) (1) (نا 164 - 21% =0 =) -5x₂=0 12=0 Put H2=0 in equation (i) => 32, + 4x(0)=0 =) 2,50 So, linear combination x (37+WV.) + (YV+7V)=0 We get n =X 1 co uvi +7V2} 2 Hence, { 30, +402 are said to be linearly independent. THANK You

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