Finding basis of a column space/row space?
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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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Answer:
No, we cannot use row as well as column operations to reduce a matrix to find column space. Consider the following counterexample: The column space of the matrix [1 1] [1 1] is the set of points on the line y = x in the coordinate plane. However, if we row-reduce the above matrix by subtracting the first row from the second row, the column space of the resulting matrix [1 1] [0 0] becomes the x-axis! Lord bless you today!
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