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Can someone explain intuitively what the Fundamental Theorem of Linear Algebra states? and why specifically it is called the above? Specifically, what makes it 'Fundamental' in the broad scope of the theory.

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    Read http://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra and the article by Strang in the Monthly cited in that page.2011-02-09
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    Argh, I rather disliked this "fundamental theorem". The whole statement in terms of orthogonality of the kernel and the co-image is obscuring the real statement about vector spaces and their duals. The fundamentality of the theorem (the "non-obvious" relation between kernel, cokernel, image, and coimage) is only due to the setting of working on $\mathbb{R}^n$ and identifying with its dual space using the standard inner product. (The other half of the theorem [rank-nullity], however, is a very fundamental one, I think. It makes the definition of dimension sensible.)2011-02-09
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    It all boils down to this: The only invariant of a finitely generated vector space is its dimension.2011-05-17
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    What's the real statement about vector spaces and their duals that @WillieWong referred to?2013-08-14
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    @littleO: the statement is relatively trivial. Let $T:V\to W$ be a linear map, it induces the [dual map](http://en.wikipedia.org/wiki/Dual_space#Transpose_of_a_continuous_linear_map) $T^*: W^* \to V^*$ where $W^*$ is the dual space of $W$. Then by definition the duality pairing $\langle T^* w^*,v\rangle = \langle w^*,Tv\rangle = 0$ for every $w^*\in W^*$ and for every $v$ such that $Tv = 0$. Or in words, "the image of $W^*$ under $T^*$ annihilates all members of the kernel of $T$".2013-08-14
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    @WillieWong Thanks, that's an interesting way of looking at it that I hadn't thought of.2013-08-14
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    Never heard of a theorem called like that. At first sight the result is even more localised than the Fundamental Theorem of Algebra. Nothing for other fields than $\Bbb R$, nothing for abstract spaces and morphisms, nothing for infinite dimensional spaces. Just curious, is there any indication of anybody using this nomenclature independently of Gilbert Strang?2013-08-14

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