TIFR-2012-Maths-B: 13

Question

True/False Question:

The polynomials $\left ( t-1 \right )\left ( t-2 \right ),\left ( t-2 \right )\left ( t-3 \right ),\left ( t-3 \right )\left ( t-4 \right ),\left ( t-4 \right )\left ( t-6 \right )\in \mathbb{R}\left [ t \right ]$ are linearly independent.

Answer 1

given polynomials    (t)2 – 3t+2   (t – 1) (t – 2)   t= 1,2  independent solutions  t = 3,4,6…..; are dependent solutions

                                 (t)2 – 5t+6    (t – 2)(t – 3)    t= 2,3  independent solutions  t = 4,6…..; are dependent solutions

                                 (t)2 – 7t+12    (t – 3)(t – 4)     t= 3,4  independent solutions  t = 6,8,…..; are dependent solutions

                                  (t)2 – 10t+24  (t – 4)(t – 6)      t= 4,6  independent solutions  t = 8,12,…..; are dependent solutions

 Therefore the above solutions are linear independent solutions for R(t).