Linear Transformation Between Different Dimension Vector Spaces

Question

If there is a linear transformation from a smaller vector space to a larger one which is 1-1 and onto (can it be)?
What will happen if the transformation is from a bigger vector space to a smaller one, some basis vector will must be sent to the $Ker(T)$ so it can not be 1-1?

Answer 1

When we talk of the 'size' of vector spaces, unless they are over a finite field, we talk of their dimension.

So we have this: Lets say T maps from V to W. lets assume dimW < dimV

we know dimV = dimImT + dimNullT > dim W

If T is onto dimNullT > dimW - dimImT = 0

then T is necessarily not one to one.

Assume dimW > dimV to get that the transformation cannot be onto..