Column Space and Pivot Columns in Reduced Matrices

horefaen
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To find the column space of a matrix, you reduce the matrix and those columns that contains leading variables(pivot columns), refers to the columns in the original matrix who span the columnspace of the matrix. But does the pivotcolumns in the reduced matrix also span the column space of the original matrix?
 
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No. Consider the following matrix:
\begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}

The row echelon form of this matrix is
\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}

Is Span\{\begin{pmatrix} 1 \\ 0 \end{pmatrix}\}= Span\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}\}?
 

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