Matrix Row Rank and Column Rank

This lecture continues the discussion on matrix row rank and column rank, showing that for equivalent row matrices A and B, their row spaces are equal. The row rank of A equals the row rank of B. The proof involves showing that each row of A is a linear combination of the rows of B, and vice versa. For a row-echelon matrix A, the row rank is equal to the number of pivots, and a basis for the row space is formed by the non-zero rows of A. Examples are provided to illustrate these concepts, along with applications to finding bases of vector spaces and completing them. The lecture concludes with practical applications and computations.