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Matrix Vector Spaces, Column Space is a subspace of <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mmultiscripts> <mtext> R </mtext> <none/> <mtext> m </mtext> </mmultiscripts> </mrow> </math>, Null Space has dimension dim(null(A))=n-r, where n is the number of columns of A and r is the rank, Column Space is defined The set of all possible linear combinations of the columns of matrix A, dim(null(A))=n-r, where n is the number of columns of A and r is the rank where Rank, Null Space e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> A= </mtext> <mfenced open="[" close="]"> <mtext> 1 2 2 4 </mtext> </mfenced> <mtext> , null(A)=span </mtext> <mfenced open="{" close="}"> <mfenced open="[" close="]"> <mtext> -2 1 </mtext> </mfenced> </mfenced> <mtext> ,dim(null(A))=1 </mtext> </mrow> </math>, dim(col(A))=r, where r is the rank where Rank, Column Space has dimension dim(col(A))=r, where r is the rank, Rank is defined as The number of non-zero rows in the reduced row echelon form of A, Matrix Vector Spaces e.g. Null Space, Column Space denoted col(A)
