Finding basis of a column space/row space

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    Basis Column Space
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SUMMARY

The discussion clarifies that both row and column operations can be utilized to reduce a matrix, but they serve different purposes in determining the column space and row space. The column space is defined as the space spanned by the columns of a matrix, while the row space is defined by the rows. It is established that the column space and row space are not generally the same, especially in non-square matrices, although they share the same dimension in square matrices, making them isomorphic.

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  • Understanding of matrix operations, specifically row and column operations.
  • Familiarity with concepts of vector spaces and their dimensions.
  • Knowledge of matrix transposition and its effects on row and column spaces.
  • Basic linear algebra principles, including the definitions of column space and row space.
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ashina14
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I was wondering whether we can use row as well as column operations to reduce a matrix to find column space? Or do we only have to perform row operations to reduce matrix in case of row space and column operations to find column space?
 
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You could, of course, take the transpose of a matrix, so that "columns" become rows and vice versa, but the "column space" of a matrix (the space spanned by its columns as vectors) is NOT, in general, the same as the "row space" of a matrix (the space spanned by it rows). If the matrix is not square, the will have completely different dimensions. If the matrix is square, the row space and column space will have the same dimension and so be "isomorphic" but not, in general, the "same" spaces.
 

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