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=x0,α2=x1,α3=x2,α4=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’={α1,α2,α3},
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’.
