Question

I have to vectors and know that they are not linearly independent. Does this also mean that they are parallel?
By vector I don´t necessarily mean an "arrow". It could also be a function or a matrix.

Answer 1

In the case of abstract vector spaces, it doesn't really make sense to say two objects are parallel if the are linearly dependent. You would just want to say that one is a scalar multiple of another. Its not until you define an inner product (like the dot product) on a space that you can start talking about an "angle" inbetween two objects, and then one can define parallel as the angle between two objects being 0.

Answer 2

Hmmmm ... okay ... I thought that parallel in vector spaces is a synonym for linearly dependent.

Answer 3

Without an inner product defined on a space, the term "parallel" is not well-defined. For example, in R^n, we have the dot product, and we know that:\[x\cdot y=|x||y|\cos \theta\]so we have a definition for the angle inbetween two objects.

But there are infinitely many inner products on R^n, each giving rise to what we could define as the angle between two objects. In one inner product, two objects might be "parallel", but in another inner product, the same two objects might be "perpendicular". This is why you dont really say parallel until youve clearly defined the inner product first.

Again, all this is about abstract vector spaces. If you are in R^n, the dot product is the standard, and so we assume it is there unless told otherwise.

Answer 4

http://en.wikipedia.org/wiki/Inner_product_space

Answer 5

Thank you, joemath314159.