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
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
consider the transposition of , , 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:
fm(r'\[1 \cdot {m1.submatrix(0,0,3,1)} + 3 \cdot {m1.submatrix(0,1,3,1)} = {m3.submatrix(0,0,3,1)}\]')
here the output vector is a linear combination of the 2 vectors and
are the coefficients
the columns of span the columns of the rows of span the rows of
given
if is an invertible matrix then:
if is an invertible matrix then
where is the matrix in its reduced echelon form
given 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