Question

Chapter 7, Section 7.3, Question 13 Determine whether the members of the given set of vectors are linearly independent for -- <t<. If they are linearly dependent, find the linear relation among them. x(1)(+) *(?)() - -602-7). «(20) = (23) | 2e-t/ linearty dependent, x(1)(t) - x2)(t) + x)(+) - 0 Iinearly dependent, x' (t) – x2)(t)-4x)(t) = 0 linearly independent linearly dependent, -2x)(1)-4x2)(t) - x)(t) - 0 linearly dependent, 2x{1}(t) - 4x2)(t) + X(t)- 0

Answer 1

Solution Given that (3) X(t) X(t) = x = 9 Now -Ked] ( ) 03 let a, b are scalars, such that 4) = xL) alt com) 6239 )+ ( a ) = 12**) ax (+) + bx = a x (t) © +6 -+ aettb -Laët bet tbet (**) (1) (a - b) et (2a+b) et and - 2a+b=0 z a b = -2 adding @ and -> a = 2 putting in - a= -2 2-5 = -2 b= 4 1 (3) Now from equation 22") + 4 of264) = since, vector xe (t) be expressed as linear combination "(t) and (4) these are linearly dependent. can a of

from equation OH) + 4 200g afect) x 60) Multiplying by minus -2x"64) - 4x = - de 6 -2x"%) - 4x0 ) + WEG finally option = linearly dependent (1) -2.0()-40 (t) of X DOO