I get always confused when these terms are mixed. Linearly independent is easy to understand. Can somebody explain in little more depth or point me to an article or book that compares these terms.
what is the difference and similarities between these terms? General position, affinely independent, linearly independent.
1 Answers
An affine combination of a bunch of vectors is a linear combination in which the coefficients add up to 1. The vectors are linearly independent if no one of them is a linear combination of the others; the vectors are affinely independent if no one of them is an affine combination of the others. If they are linearly independent then they are a fortiori affinely independent, but in ${\bf R}^2$, the vectors $(1,0),(0,1),(1,1)$ are affinely independent but not linearly independent.
As to general position, the definition depends on the context. For example, for some applications, points in the plane are said to be in general position if no three of them lie on a line; for other applications, that's not good enough, and we need to insist that no four lie on a circle.
A connection between the concepts is that three points in ${\bf R}^2$ are affinely independent if they don't all lie on the same line, which is the same as the looser definition of general position.