# Column space vs. Span

• Aug 27th 2012, 07:31 AM
phys251
Column space vs. Span
Can someone please refresh me on the difference between what Col A and Span A are?
• Aug 27th 2012, 09:43 AM
HallsofIvy
Re: Column space vs. Span
Span A exists only if A is a set of vectors. It is the space of all vectors that can be written as a linear combination of vectors in A.

Col A exists only if A is a matrix- it is the space spanned by the columns of A thought of as separate vectors.
• Aug 27th 2012, 03:50 PM
Deveno
Re: Column space vs. Span
they are related, like so: an mxn matrix A can be thought of a collection of n column vectors:

A1,....,An, where each Aj is an m-vector (an mx1 column vector).

in which case, Col(A) = Span({A1,....,An}).

note that although a set doesn't need to be "in order", re-arranging the columns gives a different matrix. however, you will still get the same column space.

that is, if we re-arrange the columns to get A', Col(A) = Col(A') (because Span(S) is the span of a SET S, and sets don't care what order they're listed in).

it is (somewhat) important to draw a distinction between A and {Aj: j = 1,2,....,n}. the collection of n mx1 column vectors is a set with n things in it. the matrix A is but a single object.