6

I'm studying linear algebra and all of a sudden the symbol $\dot{+}$ appears. For example: $a*(v \dot{+} w) = a*v \dot{+} a*w$ Any idea what it might be?

Also two more symbols. they are on top of $0$ in equations: - and ~. For example $u \dot{+} w = \bar{0}$ and $\tilde{0}=\bar{0} + \tilde{0} = \bar{0}$.

Sorry if those questions are stupid but let's just say that linear algebra is not my favourite subject that I take in university

David Richerby
  • 2,450
RedRose23
  • 67
  • 2
    While the current answer is a likely explanation, you won't get a real answer without more context. Are you reading this from a textbook? Could you take a picture of a page featuring these symbols (the entire page, not just the symbols). – Jack M Feb 28 '15 at 18:08
  • These are not standard symbols so I guarantee that the book will define them where they're first used. – David Richerby Feb 28 '15 at 22:56

1 Answers1

19

It seems that the first is a "decoration" of the usual symbol just used to highlight the difference between this (abstract) operation in a vector-space and a "usual" addition in the real number say. The context is not quite sufficient to be absolutely sure though.

But one could have something like this: for $v,w\in V$ and $a,b \in \mathbb{R}$ one has $a(v \dot{+}w)= av \dot{+} a w$ and $(a+b)v = av \dot{+} bv$; note the usual plus, is in the real numbers the one with the dot in the vector-space.

For the second it is more clear what is happening. Every vector-space (or every additive group) has a "zero-element" that is an element that is neutral with respect to the additive operation. One could also just denote this $0$ but one might use $\overline{0}$ to distinguish it from the $0$ in the real numbers.

For example one can then say: for each $v \in V$ one has $0v= \overline{0}$; note the usual zero is the real number and we say that scalar multiplication by the real number $0$ yields the zero-element of the vectorspace.

Finally the two different "decorations" are almost certainly part of a proof that the zero-element is unique. So one assume there are two elements $\overline{0}$ and $\tilde{0}$ that behave like a zero-element and then shows they are equal.

Summary: it is common to modify common symbols, like $+,0,1$ for keeping the intuition conveyed by the usual symbol while not using the exact same symbol for different things.

quid
  • 42,135
  • 2
    I like your answer. At first I was thinking of $\dotplus$ in the context of direct sums, and then putting hats on things to denote equivalence classes in a quotient space, but this seems far more likely. – pjs36 Feb 28 '15 at 18:04
  • 5
    Yeah those are the exact symbols and few of the equations that the book has. You explained it perfectly! Thanks you rock! – RedRose23 Feb 28 '15 at 18:11
  • I remember being in a first year linear algebra class and the professor did something similar, so that students wouldn't take for granted any algebraic properties we were used to from high school. – eigenchris Feb 28 '15 at 23:13