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Rank Theorem



  • For a `mtimes n` matrix `A`, it maps a point in `n`-dimension to one in `m`-dimension.
\begin{align} \dim \ker A + \dim \text{Im } A = \dim \ker A + \operatorname{rank} A = n
\end{align}

   Original `n`-dimension is suppressed by`dim text{ker}  A` and mapped by`dim text{Im}  A =`` text{rank}  A` to the objective space.
  • If `m>n`, the mapping by `A` is to the higher dimension. However, it cannot cover that dimension all.
  • If `m<n`, the mapping by `A` is to the lower dimension. So, many points map to one point, which means a kind of suppresion.
  • Let `Ax=y`. For `x_1` and `x_2`, if `x_1=x_2` when `y=Ax_1=Ax_2`, then `A` is injective. If there exists `x` such that `y=Ax` for all `y`, then `A` is surjective.
  • `text{rank}  A leq m`. It means that the objective space is `m`-dimensional, so `dim text{Im}  A` is at most `m`. 
  • `text{rank}  A leq n`. It means that the original space is `n`-dimensional, so `dim text{Im}  A` is at most `n` although it covers the objective space all.        


︎Reference

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

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