Chapter 2.4 (Linear Dependence and Span), page 36 of Deep Learning by Goodfellow, Bengio, and Courville, claims the following:
No set of $m$-dimensional vectors can have more than $m$ mutually linearly independent columns, but a matrix with more than $m$ columns may have more than one such set.
I understand that no set of $m$-dimensional vectors can have more than $m$ mutually linearly independent columns, but I'm unsure of what the latter claim is trying to say.
It's probably the way that this is phrased that is confusing me, so I would appreciate it if people could please take the time to clarify this.
Best Answer
A 4x8 matrix may have many sets of 4 linearly independent columns. Perhaps the first 4 columns are independent, and so are the last 4 columns