That doesn't seem right to me. The matrix just can't be wider than it is tall (which is already implied by requiring the columns to be independent). Are you sure we're not also requiring something else, like surjectivity?
In my linear algebra book, they state that the only way for the null space of a matrix to contain just the zero-vector, the column vectors of the matrix has to be linearly independent, and the matrix has to be a square matrix.
Now, I understand why the column vectors of the matrix have to be linearly independent, but why does it have to be a square matrix?
Would appreciate any explanation!
I agree with Tinyboss, your book is completely wrong. Unless you overlooked something in the statement, I would get another book if I were you. What book is it?
I would suggest this book: http://www.amazon.com/Linear-Algebra.../dp/0130084514.
i can personally recommend this text:
Amazon.com: Linear Algebra (2nd Edition) (9780135367971): Kenneth M Hoffman, Ray Kunze: Books