Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m × n matrix A and its singular value decomposition:

First, each matrix ${\displaystyle A\in \mathbb {R} ^{m\times n}}$ (${\displaystyle A}$ has ${\displaystyle m}$ rows and ${\displaystyle n}$ columns) induces four fundamental subspaces. These fundamental subspaces are:

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