please don\'t rush Is the set os vectors S =n {1 - x, 1 - x2, x - x}P2 linearly
ID: 2832180 • Letter: P
Question
please don't rush
Is the set os vectors S =n {1 - x, 1 - x2, x - x}P2 linearly dependent or linearly independent? Show all your work for full marks. The three binomials are linearly dependent because one can be matter as a linear combination or the other two 1 - X = 1 * (1 - X) - 1 * (X - X2) clearly one is a linear dependent Also, we can say that x - X2 can be Represented as (1 - X2) - (1 - X) so they are linearly dependent as they can be expressed with the help of other two vectors a(1 - x) + b(1 - x)(1 + x) + c(x)(1 - x)(1 + x) a-ax + b - bx2 + cx - cx2 a + b + (c - a)x - (b + c)x2 = 0 eor this to be zero, c = d b = -c a = -b Here, the only solution that exists is a = b = c = 0 Thus, this is linearly independent.Explanation / Answer
Two vectors u and v are linearly independent if the only numbers x and y satisfying xu+yv=0 are x=y=0.
so let there exist a,b,c such that
a(1 - x) + b(1 - x2) + c(x - x2) = 0
=> a + b - ax + cx -bx2 - cx2 = 0
=> a + b + x(c - a) - x2(b + c) =0
=> x2(b + c) - x(c - a) - (a + b) =0
=> discriminant of above quadratic equation
D = (c-a)2 + 4*(b+c)*(a + b)
=> D > 0 when a , b , c>0
Hence there may exist a solution to above equation
Hence the given set of vectors are linearly dependent.
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