I have been reviewing criteria for bijectivity of linear mappings because I am going to have to be able to proof similar results on an upcoming exam and use various theorems to deduce if a linear transformation is one-one, onto, ect. One theorem I am not so sure how to prove is below. I was wondering if it came up in a standard textbook on linear algebra because it was presented without proof in a set of lecture notes I am using.
Let $V,W$ be vector spaces over some field $F$ and suppose $f$ is a linear transformation between these spaces.
How do we show $f:V\rightarrow W $ is one to one if and only $dim(V) \leq dim(W)$ and if A is the matrix corresponding to the linear transformation f(with respect to the standard basis of V and W) then there exists no scalar $c \neq 0$ such that the products by $c$ of all minors of order $dim(V)$ of $A$ are zero.