I am trying to study lineair algebra on my self, and I came across a weird phenomenom "abusing a notation" What do they mean by that , does that mean that it isnt mathematical correct? Or is it a connection that doesn't really have meaning?
Abusing notation ? What does that mean
linear algebranotation
Related Solutions
Let $S$ be a set of (logical) formulae and $\psi$ be a formula. Then $S \vdash \psi$ means that $\psi$ can be derived from the formulae in $S$. Intuitively, $S$ is a list of assumptions, and $S \vdash \psi$ if we can prove $\psi$ from the assumptions in $S$.
$\vdash \psi$ is shorthand for $\varnothing \vdash \psi$. That is, $\psi$ can be derived with no assumptions, so that in some sense, $\psi$ is 'true').
More precisely, systems of logic consist of certain axioms and rules of inference (one such rule being "from $\phi$ and $\phi \to \psi$ we can infer $\psi$"). What it means for $\psi$ to be 'derivable' from a set $S$ of formulae is that in a finite number of steps you can work with (i) the formulae in $S$, (ii) the axioms of your logical system, and (iii) the rules of inference, and end up with $\psi$.
In particular, if $\vdash \psi$ then $\psi$ can be derived solely from the axioms by using the rules of inference in your logical system.
I agree with the commenters: it's most likely the linear span ($L$ for linear) of the vectors in parenthesis. But that is from context rather than from convention.
In general, the best place to look for the answer to a question like this is the materials you are studying from.
When you take a course, you get a textbook and/or lecture notes. Those texts are designed to be coherent and consistent. The definitions and notations may vary from text to text. Notation is calculus has pretty much been standardized, but less so in linear algebra.
When you're self-studying, I think you need to work in the same way. Concentrate on your one source. If there is a definition or notation you don't understand, work backwards until its first usage. There you'll find its meaning.
Best Answer
Strictly speaking, if $f(z) = z^k$, then the function is $f$, not $z^k$. But going through the contortions to add another layer of notation would only obscure the plain meaning. So since "we all know" that by $z^k$, we mean the function $z \mapsto z^k$, then let's just write $z^k$.
This depends on what definition for "function" your text actually uses, so I'm guessing a little bit here.