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If V and W are finite-dimensional vector spaces, and dim(V) does not equal dim(W) then there is no bijective linear transformation from V to W.

An isomorphism between V and W is a bijective linear transformation from V to W. That is, it is both an onto transformation and a one to one.

1- Question

Let W be a proper subset of an vector space V, and let T be the projections onto W. Prove that T is not an isomorphism.

2- Answer

Since T is a projection onto W then, T(v)=w, therefore dim(V) > dim(W)

However since W is a proper subset of the vector space V, W is missing an element of V and therefore dim (w) is smaller then dim (V) and is not equal. Therefore it is not a one to one, and so is not an isomorphism.

This is my answer, however I am unsure if that is correct. It makes sense in my mind that a subset of a vector space is smaller then the vector space itself in dimension, however I am uncertain. Can you help me please?

Thank you,

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# Linear Algebra; Isomorphism

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