1) For the set of vectors S = {x,.x,}.. where X, = (1,3),X, =(4,6) are in...

   

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1) For the set of vectors S = {x,.x,}.. where X, = (1,3)",X, =(4,6) are in IR find the matrix of the linear transformation T: IR? → IR, such that TX, =(-2,2,-7) and TX, = (- 2,-4,-10). Ans:-4 2 2 -3 2x+ 2) Define two functions T: IR? → IR' and S: IR? → IR? by T OF) Determine whether T, S, and the composite SoTare linear transformations. Ans: Tand SeT are linear transformations, but Sis not. 3) Let Vdenote the vector space of 2x2 matrices, and WW the vector space of 3x2 matrices. Define [a+b 2b -d -3c. Find a basis for the 2d the linear transformation T:V → W by 2 |2b -с - За ] range of T. [[1 0 1 0][0 o Го Ans: Basis ofT ={0 0 2 0.0 -3-1 -1 0 0 -3||2 4) Let T be the linear transformation from the 3-dimensional vector space IR to IR itself satisfying the following relations: T Then for any vector xe IR, find the formula for T(x). [-3x +7y- 3z] Ans: T(x)= 2x-4y+2z L-2х +бу-3г 5) Let T: IR → IR' be a linear transformation such that T Then find the matrix A such that T(x)=Ax for every XEIR?, and find the rank and nullity of T. Ans: A = Rank(T)-2, Nullity(T)=0. -1 2 1) For the set of vectors S = {x,.x,}.. where X, = (1,3)",X, =(4,6) are in IR find the matrix of the linear transformation T: IR? → IR, such that TX, =(-2,2,-7) and TX, = (- 2,-4,-10). Ans:-4 2 2 -3 2x+ 2) Define two functions T: IR? → IR' and S: IR? → IR? by T OF) Determine whether T, S, and the composite SoTare linear transformations. Ans: Tand SeT are linear transformations, but Sis not. 3) Let Vdenote the vector space of 2x2 matrices, and WW the vector space of 3x2 matrices. Define [a+b 2b -d -3c. Find a basis for the 2d the linear transformation T:V → W by 2 |2b -с - За ] range of T. [[1 0 1 0][0 o Го Ans: Basis ofT ={0 0 2 0.0 -3-1 -1 0 0 -3||2 4) Let T be the linear transformation from the 3-dimensional vector space IR to IR itself satisfying the following relations: T Then for any vector xe IR, find the formula for T(x). [-3x +7y- 3z] Ans: T(x)= 2x-4y+2z L-2х +бу-3г 5) Let T: IR → IR' be a linear transformation such that T Then find the matrix A such that T(x)=Ax for every XEIR?, and find the rank and nullity of T. Ans: A = Rank(T)-2, Nullity(T)=0. -1 2