Would it be wrong to write the span with redundancies?
ericpan64
To my knowledge, bases shouldn't have redundancies but I'm not too sure if the same applies to a span. Also, what is the distinction between a basis and a span? They seem pretty similar to me Thanks!
TLindberg: May 5, 2015, 12:05 a.m.
Technically it wouldn't be incorrect for a span to have redundancies as far as I know, but those vectors would not form a basis as they are not linearly independent because a redundant vector is a scaling of another vector in the set, causing it to be a linearly dependent set. The requirements of a basis set are that the set of vectors: 1. Span the vector space which they represent 2. Are linearly independent of each other Therefore you could get a set of vectors that does span a vector space but is not a basis set for that vector space, because the vectors are not linearly independent. In general if they are asking for a set of vectors that span a vector space or a subspace I would say it's for the best to eliminate redundancies or linearly dependent vectors. The span will then show the dimension of the vector space by its number of vectors, which could be another question or part asked later on.
TLindberg: May 5, 2015, 12:20 a.m.
Sorry, the formatting got a bit messed up above. Hopefully this works, if not sorry for spamming your question with the same comment. \\ Technically it wouldn't be incorrect for a span to have redundancies as far as I know. However those vectors would not form a basis as they are not linearly independent because a redundant vector is a scaling of another vector in the set, causing it to be a linearly dependent set. \\ The requirements of a basis set are that the set of vectors: \\ 1. Span the vector space which they represent \\ 2. Are linearly independent of each other \\ Therefore you could get a set of vectors that does span a vector space but is not a basis set for that vector space, because the vectors are not linearly independent. \\ In general if they are asking for a set of vectors that span a vector space or a subspace I would say it's for the best to eliminate redundancies or linearly dependent vectors. The span will then show the dimension of the vector space by its number of vectors, which could be another question or part asked later on.
brian_li: May 6, 2015, 6:24 p.m.
A span should not have redundant vectors. When it asks to find a set of vectors that "span" the image or kernel, its basically asking to find the smallest set of independent vectors of the image or kernel.