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Linear Independence, Dimension, Span and Basis, Suppose A is an mxn matrix with rank(A)=r, where r<min(m,n) with m>n 1. The columns of A are linearly dependent 2. The rows of A are linearly dependent 3. dim(null(A))=n-r 4. dim(col(A))=r 5. dim(row(A)=r 6. There are m-r zero rows in rref(A), Linear System of Equations which can be represented mathematically as Ac=b, Span e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mfenced open="{" close="}"> <mfenced open="[" close="]"> <mtext> 1 3 </mtext> </mfenced> <mtext> , </mtext> <mfenced open="[" close="]"> <mtext> -2 -2 </mtext> </mfenced> <mtext> , </mtext> <mfenced open="[" close="]"> <mtext> -1 1 </mtext> </mfenced> </mfenced> </mrow> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mfenced open="{" close="}"> <mfenced open="[" close="]"> <mtext> 1 0 0 </mtext> </mfenced> <mtext> , </mtext> <mfenced open="[" close="]"> <mtext> -1 0 4 </mtext> </mfenced> <mtext> , </mtext> <mfenced open="[" close="]"> <mtext> 0 2 0 </mtext> </mfenced> </mfenced> </mrow> </math> e.g. This set of vectors has dimension 3, they span a 3-dimensional subspace which happens to be all of R^3, Rank e.g. Suppose A is an mxn matrix and rank(A)=m, a whole number less than or equal to the number of components in each of the vectors which are elements of the vector space such as This set of vectors has dimension 3, they span a 3-dimensional subspace which happens to be all of R^3, a whole number less than or equal to the number of components in each of the vectors which are elements of the vector space such as This set of vectors has dimension 2, they span a 2-dimensional subspace which happens to be all of R^2, A set of vectors that are linearly independent and span the vector space in question the number of vectors in a basis for a vector space is the definition of Dimension, Dimension is always a whole number less than or equal to the number of components in each of the vectors which are elements of the vector space, Linear Combination is defined as <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> The vector </mtext> <mmultiscripts> <mtext> c </mtext> <mtext> 1 </mtext> <none/> </mmultiscripts> <mmultiscripts> <mtext> v </mtext> <mtext> 1 </mtext> <none/> </mmultiscripts> <mtext> + </mtext> <mmultiscripts> <mtext> c </mtext> <mtext> 2 </mtext> <none/> </mmultiscripts> <mmultiscripts> <mtext> v </mtext> <mtext> 2 </mtext> <none/> </mmultiscripts> <mtext> +...+ </mtext> <mmultiscripts> <mtext> c </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mtext> v </mtext> <mtext> n </mtext> <none/> </mmultiscripts> </mrow> </math>
