**Rank Theorem**

- For a `mtimes n` matrix `A`, it maps a point in `n`-dimension to one in `m`-dimension.

\end{align}

- 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.

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[1] Hiraoka Kazuyuki, Hori Gen,

**Reference**[1] Hiraoka Kazuyuki, Hori Gen,

*Programming No Tame No Senkei Daisu*, Ohmsha.emoy.net