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Mathematics-2014: Answer Writing Challenge – 5

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19 September 2014

 

1) Let T be a linear operator on V3(R)  defined by

T (a,b,c)=(3a,a-b, 2a+b+c)  for all (a,b,c)єV3(R).Is T invertible? If so, find a rule for T-1 like the one which defines T.

 

2) Let V(R) be the vector space of all polynomials in x with coefficients in R of the form

f(x)=a0x0+a1x+a2x2+a3x3   i.e.,the space of polynomials of degree three or less.The differential operator D is a linear transformation on V.The set B={α1,….,α4} where α1=x02=x13=x24=x3

is an ordered basis of V. Write the matrix of D relative to the ordered basis B.

 

3) Let T be the linear operator on R3 defined by

T(a,b,c)=(3a+c, -2a+b, -a+2b+4c).

  • What is the matrix of T in the standard ordered basis B for R3?
  • Find the transition matrix p from the ordered basis B to the ordered basis B’={α123},

where α1=(1,0,1), α2=(-1,2,1),and α3=(2,1,1).Hence find the matrix of T relative to the ordered basis B’.