Can anyone explain relationship between "onto" and "columns are independent" ?

Question

I remember reading this statement before.

It is as follows.

Transformation is onto if and only if columns are linearly independnet

Transformation is one-to-one if and only if rows are independent

I think it is not right statement because it comes from my unclear memory of reading this

statement before.

But, what I read is quite similar to above statements, but can't recall perfectly.

Can anyone please modify it ?

Answer 1

You've just mixed up your two theorems a bit. Here are the correct statements:

$\textbf{Theorem: }$ Let $T:\mathbb{R}^n \rightarrow \mathbb{R^m}$ be a linear transformation, and let $A$ be the standard matrix of $T$. Then:

1. $T$ is one-to-one if and only if the columns of $A$ are linearly independent.
2. $T$ is onto if and only if the columns of $A$ span $R^m$.

I recommend that you look at Is a linear tranformation onto or one-to-one? for the full proof of this theorem, and for additional theorems related to the topic.