Rank Theorem

  • For an m×nm \times n matrix AA, it maps a point in nn-dimension to one in mm-dimension.
dimkerA+dimImA=dimkerA+rankA=n\begin{aligned} \dim \ker A + \dim Im A = \dim \ker A + rank A = n \end{aligned}

Ratio

Original nn-dimension is suppressed by dimkerA\dim \ker A and mapped by dimImA=rankA\dim Im A = rank A to the objective space.

  • If m>nm > n, the mapping by AA is to the higher dimension. However, it connot cover that dimension all.
  • If m<nm < n, the mapping by AA is to the lower dimension. So, many points map to one point, which means a kind of suppresion.
  • Let Ax=yAx = y. For x1x_1 and x2x_2, if x1=x2x_1 = x_2 when y=Ax1=Ax2y = Ax_1 = Ax_2, then AA is injective. If there exists xx such that y=Axy = Ax for all yy, then AA is surjective.
  • rankAmrank A \le m. It means that the objective space is mm-dimensional, so dimImA\dim Im A is at most mm.
  • rankAnrank A \le n. It means that the original space is nn-dimensional, so dimImA\dim Im A is at most nn although it covers the objective space all.

Volume

Reference

[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.


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