A set of vectors spans a space if every other vector in the space can be written as a linear combination of the spanning set. But to get to the meaning of this we need to look at the matrix as made of column vectors.
Here's an example in #mathcal R^2#:
Let our matrix #M = ((1,2),(3,5))#
This has column vectors: #((1),(3))# and #((2),(5))#, which are linearly independent, so the matrix is non-singular ie invertible etc etc.
Let's say that we want to show that the generalised point #(x,y)# is within the span of these 2 vectors, ie so that the matrix spans all of #mathcal R^2#, then we look to solve this:
#alpha ((1),(3)) + beta ((2),(5)) = ((x),(y))#
Or:
#((1,2),(3,5)) ((alpha),(beta))= ((x),(y))#
You can solve this is any number of ways, eg row reduce or invert M.....to get:
#alpha = - 5x + 2y, beta = 3x - y#
So let's say we want to check that #(2,3)# is in the span of this matrix, M, we apply the result we just got:
#alpha = -4#
#beta = 3#
Double-check:
#-4 ((1),(3)) + 3 ((2),(5)) = ((2),(3))# !!
Consider next a different matrix: #M' = ((1,2),(2,4))#. This is singular because its column vectors, #((1),(2))# and #((2),(4))#, are linearly dependent. This matrix only spans along direction #((1),(2))#.
