# Can anyone explain relationship between “onto” and “columns are independent” ?

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 ?

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.

• For the second statement, How can columns of A span R^m? Matrix of A consists of n columns,, – user197971 Dec 10 '14 at 4:12
• As you said, A will consist of n columns. These column vectors will be elements of R^m. Thus, a necessary condition for the columns of A to span R^m is that n is greater than or equal to m. That is, if n<m, then the columns of A certainly do not span R^m, and T is therefore not onto. Does that answer your question? – Gecko Dec 10 '14 at 4:50
• certainly thanks! – user197971 Dec 10 '14 at 5:23
• You're welcome, glad I could help! – Gecko Dec 10 '14 at 6:16