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January 29th, 2010, 12:41 PM   #1
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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?

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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.
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January 29th, 2010, 11:51 PM   #3
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Re: linear transformation

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