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Prove that if $A$ is an $m \times n$ matrix over the field $\mathbb R$ of real numbers, then $\mathbb R^n = \operatorname{Col}(A^T) \oplus N(A),$ where $\operatorname{Col}(A^T)$ denotes the column space of $A$ and $N(A)$ denotes the right null space of A.

Dylan C. Beck
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Gunjan ojha
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  • Please give some completments about the idea you have. I once have been told this many times... – Mod.esty May 26 '20 at 16:23
  • hint: consider the inner product and orthogonal complement subsapce of $N(A)$... – Mod.esty May 26 '20 at 16:30
  • Essentially, this is the Rank-Nullity Theorem. Can you emulate the proof for that? – Dylan C. Beck May 26 '20 at 16:45
  • @Carlo Yes, it is right over any field. – Mod.esty May 26 '20 at 16:55
  • This question has been asked many times. Here is one where I posted an answer: https://math.stackexchange.com/questions/21144/intuitive-explanation-of-the-fundamental-theorem-of-linear-algebra Here's another thread: https://math.stackexchange.com/questions/318136/the-range-of-t-is-the-orthogonal-complement-of-kert – littleO May 26 '20 at 17:29

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