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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 bydim text{ker}  A and mapped bydim 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

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

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