How are the column space and the row space of a matrix A related to the...

How are the column space and the row space of a matrix A related
to the column space and row space of its reduced row echelon form?
How does this prove the column rank of A equals the row rank?

Solutions

Expert Solution

A row operation, either interchanges 2 rows or multiplies a row by a scalar or adds a scalar multiple of a row to another row. Therefore, row operations create scalar multiples or linear combinations of rows of a matrix. Thus, if the matrices A and B are related by elementary row operations, then Row(A) is same as Row(B), i.e. the row spaces of the 2 matrices A and B are same.

However, the same cannot be said about the column spaces of the 2 matrices A and B, which may or may not be related to each other. The row operations may change the column space. The only statements that we can make with certainty are that the row operations do not change the column rank of a matrix and that the columns bearing the same numbers which are linearly independent in the matrix A, are linearly independent in the matrix B.

This proves that the column rank of A equals the row rank as both remain unchanged by row operations.

milcah answered 1 year ago

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