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January 30th, 2010, 12:49 AM  #1 
Senior Member Joined: Nov 2009 Posts: 129 Thanks: 0  linear transportation: onetoone and onto
T,U: V ? V by T(a1, a2,....) = (a2, a3,....) and U(a1,a2,...) =(0,a1,a2...) T and U are called left shift and right shift operators on V, respectively. a) Prove that T is onto, but not onetoone b)Prove that U is onetoone, but not onto. thanks 
January 30th, 2010, 02:56 AM  #2 
Newbie Joined: Jan 2010 Posts: 4 Thanks: 0  Re: linear transportation: onetoone and onto
I assume V does not have a finite dimension. We have to show that every vector in V is in the range of T. Let Alpha = (a1, a2, a3, ...) be a vector in V. Let Beta = (a1, a1, a2, a3, ...) be a vector in V. T(Beta) = Alpha > Alpha is in the range of T. In order to show that T is not onetoone we can show that there is a non zero vector which is mapped by T to zero. Since T is a left shift we have T(x, 0, 0, 0, ...) = (0, 0, 0, ...) so T is not onetoone. In order to show that U is onetoone we can show that the only vector which is mapped by U to 0 is the zero vector. Equivalently we can show that if Alpha is not zero then U(Alpha) is not zero either. Let us assume there is a non zero vector (a1, a2, a3, ...) we'll refer to it as Alpha. if it is a non zero vector then there is an i such that ai is not zero. Since U is a right shift the (i + 1) position in U(Alpha) is not zero. Since U(a1, a2, a3, ...) = (0, a1, a2, a3, ...) all the vectors in the range of U have a zero in the first position. So there is no Alpha in V such the U(Alpha) = (1, 1, 1, 1, ...). Hope it helps... 
January 30th, 2010, 03:50 PM  #3 
Senior Member Joined: Nov 2009 Posts: 129 Thanks: 0  Re: linear transportation: onetoone and onto
thanks


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linear, onetoone, transportation 
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