Row equivalent matrices have the same column rank

Let A and B be row equivalent matrices over a field, say by an invertible matrix P such that PA = B.

  1. Prove that, for all matrices X (such that the dimensions match), AX = 0 if and only if BX = 0.
  2. Prove that any linear dependence satisfied by the columns of A is also satisfied by the columns of B.
  3. Conclude that A and B have the same column rank.

  1. If AX = 0, then PAX = P0, so BX = 0. Conversely, if BX = 0, then P^{-1}BX = P^{-1}0, so AX = 0.
  2. Write A and B as column matrices: A = [A_1 | \cdots | A_n] and B = [B_1 | \cdots | B_n]. Note that, since PA = B, we have [PA_1 | \cdots | PA_n] = [B_1 | \cdots | B_n]. In particular, PA_i = B_i for all i. Now suppose \sum \alpha_i A_i = 0 is a linear dependence among the columns of A. Now apply P (as a linear transformation) to this equation; we have 0 = P(\sum \alpha_i A_i) = \sum \alpha_i PA_i = \sum \alpha_i B_i. Thus the columns of B satisfy the same linear dependence.
  3. Suppose \{A_i\}_K is a maximal linearly independent set of columns of A. By part (2), \{PA_i\}_K = \{B_i\}_K is a linearly independent set of columns of B. If this set is not maximal, then by part (2) again (using P^{-1}) \{A_i\}_K is not maximal, a contradiction. So \{B_i\}_K is a maximal linearly independent set. Since P is injective (as a linear transformation), these sets have the same cardinality. In particular, A and B have the same column rank.
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