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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

Quote:
 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

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