Please see Attachment Respond in PDF format© BrainMass Inc. brainmass.com September 19, 2018, 4:57 am ad1c9bdddf - https://brainmass.com/math/linear-transformation/transformation-vector-spaces-31490
Please see attached file.
If A is 1 to 1, i.e., Ax=0 if and only if x=0, then the columns of A are linearly independent by definition.
To show this in more details, let's look at the following:
The Definition of ...
This looks at whether vector spaces are one-to-one or onto.