column space
the span of the columns of a matrix 
, denoted by 
, is called the range or the column space of the matrix. the row space and the column space of a matrix always have the same dimension
generally when referring to a span of a matrix we refer to the column space
here we are basically looking at the matrix as a row of columns where each column represents a vector
, denoted by , is called the range or the column space of the matrix. the row space and the column space of a matrix always have the same dimensiongenerally when referring to a span of a matrix we refer to the column space
here we are basically looking at the matrix as a row of columns where each column represents a vector
please note that elementary row operations change the column space (unlike the row space which doesnt change) so after reducing a matrix you would have to go back to the original matrix and pick the corresponding columns from there not from the matrix you applied row operations to
let 
then 
and 
then and consider the transposition of , 
, we know 
and 
, , we know and we can describe matrix multiplication using vectors and the concept of linear combinations
if
then the column 
(column 
of C) is a linear combination of the columns of using 
as the set of coefficients
you might notice that:
src_sage{fm(r'
')} {{{results(
)}}}
here the output vector is a linear combination of the 2 vectors and
are the coefficients
if
then the column (column of C) is a linear combination of the columns of using as the set of coefficients m1 = matrix([[3,1], [2,0], [1,2]])
m2 = matrix([[1,0,1], [3,2,1]])
m3 = m1 * m2
fm(r'\[{m1} \cdot {m2} = {m3}\]')
:results:
:end:you might notice that:
src_sage{fm(r'
')} {{{results()}}}here the output vector is a linear combination of the 2 vectors and
are the coefficientsthe columns of span the columns of 
the rows of 
span the rows of
span the columns of the rows of span the rows of given 
if 
is an invertible matrix then:
is an invertible matrix then:if is an invertible matrix then
where is the matrix in its reduced echelon form
is an invertible matrix then where is the matrix in its reduced echelon formgiven the matrix
let 
and let 
be a basis of 
we construct a matrix
we know 
there exists
such that 
because the rows of are spanned by the rows of
therefore
we substitute inplace of 
in the lemma and we get:
therefore
and let be a basis of we construct a matrix
we know there exists
such that because the rows of are spanned by the rows of thereforewe substitute inplace of in the lemma and we get:therefore