When will be this surjective or injective?

• Nov 27th 2011, 05:49 AM
gotmejerry
When will be this surjective or injective?
Let $\displaystyle \bold{A}$ be an $\displaystyle m \times n$ matrix and $\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ :$\displaystyle \bold{x} \rightarrow \bold{A} \cdot \bold{x}$ What is a necessary and sufficient condition for f to be injective. And surjective?
• Nov 27th 2011, 06:54 AM
Also sprach Zarathustra
Re: When will be this surjective or injective?
Quote:

Originally Posted by gotmejerry
Let $\displaystyle \bold{A}$ be an $\displaystyle m \times n$ matrix and $\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ :$\displaystyle \bold{x} \rightarrow \bold{A} \cdot \bold{x}$ What is a necessary and sufficient condition for f to be injective. And surjective?

I'ii give you the answer, and you will try to explain me why is that...

$\displaystyle f$ injective iff $\displaystyle rank(A)=n$

$\displaystyle f$ surjective iff $\displaystyle rank(A)=m$
• Nov 27th 2011, 08:04 AM
gotmejerry
Re: When will be this surjective or injective?
Can you give me a bit more hint where to start?
• Nov 27th 2011, 10:20 AM
gotmejerry
Re: When will be this surjective or injective?
For injectivity I did. It is injectien for every f(x)=f(y) ---> x=y.

$\displaystyle \begin{bmatrix} {a_{11}}\\{a_{21}}\\ \vdots\\{a_{m1}\end{bmatrix}(x_1 -y_1)+\begin{bmatrix} {a_{12}}\\{a_{22}}\\ \vdots\\{a_{m2}\end{bmatrix}(x_2-y_2)+\hdots + \begin{bmatrix} {a_{1n}}\\{a_{2n}}\\ \vdots\\{a_{mn}\end{bmatrix}(x_n-y_n)=0}$ So if the rank is n it can only be zero if $\displaystyle \bold{x}=\bold{y}$

What should i do with the surjective part?