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 January 29th, 2010, 12:41 PM #1 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 linear transformation Let V and W be finite-dimensional vector spaces and T:V ? W be linear. Prove that if dim(V) < dim(W), then T cannot be onto. I have an idea that Assume dim(v) = m and dim(W) = n. Suppose W = {w1, w2, ... wn) and V = {v1, v2, ... vm). Vectors in W will maps onto all vectors in V however, one vectors in W will not get map onto V since m > n. Therefore T is not onto. Is this right? if so how do I reconstruct it? thanks. January 29th, 2010, 04:44 PM   #2
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Re: linear transformation

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 Originally Posted by tinynerdi Let V and W be finite-dimensional vector spaces and T:V ? W be linear. Prove that if dim(V) < dim(W), then T cannot be onto. I have an idea that Assume dim(v) = m and dim(W) = n. Suppose W = {w1, w2, ... wn) and V = {v1, v2, ... vm). Vectors in W will maps onto all vectors in V however, one vectors in W will not get map onto V since m > n. Therefore T is not onto. Is this right? if so how do I reconstruct it? thanks.
Let v1,...,vm be a basis for V. Then (T(v1),...,T(vm)) will be a basis for T(V), which is therefore m dimensional (or less). There are then vectors which are not in W, since dim(W) > m. January 29th, 2010, 11:51 PM #3 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: linear transformation thanks Tags linear, transformation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Housslv Linear Algebra 0 June 13th, 2013 09:31 AM bilano99 Algebra 3 March 31st, 2012 09:43 AM richardsim10 Linear Algebra 0 February 17th, 2012 07:44 PM tinynerdi Linear Algebra 3 February 2nd, 2010 03:09 AM ypatia Linear Algebra 2 March 5th, 2009 01:49 PM

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