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Mathematics
Linear and Abstract Algebra
What is the Corollary of the Nucleus and Image Theorem?
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[QUOTE="Portuga, post: 6457648, member: 427847"] [B]TL;DR Summary:[/B] Consider ##u, v \in \mathbb{R}^2## such that ##\left\{ u,v \right\}## is a ##\mathbb{R}^2## basis. If ##F: \mathbb{R}^2 \rightarrow \mathbb{R}^n## a linear transformation, show that one of the following alternatives verifies: (a) ##\left\{ F(u), F(v) \right\}## is linearly independent. (b) ##\dim \text{Im} (F) = 1## (c) ## \text{Im} (F) = {o}## I tried hard to understand what this author proposed, but I feel like I failed miserably. My attempt of solution is here: Item (a) is verified in the case where ##n = 2##, since ##F## being a linear transformation, by the Corollary of the Nucleus and Image Theorem, ##F## takes a basis of ##\mathbb{R}^2## to a ##\mathbb{R}^2## basis. Thus, ## \left \{ F (u), F (v) \right\}## is a basis of ##\mathbb{R}^2##. Item (b) occurs in the case where ##n = 1##, because ##\dim \mathbb{R}^2 = 2, \dim \mathbb{R} = 1##, and by the Nucleus and Image theorem, $$ \begin{align*} & \dim\mathbb{R}^{2}=\dim\ker\left(F\right)+\dim\mathbb{R}\\ \Rightarrow & 2=\dim\ker\left(F\right)+1\\ \Rightarrow & \dim\ker\left(F\right)=2-1=1. \end{align*} $$ Item (c) is verified in case ##n = 0##. I am pretty sure that I am very far away of the author's intention with this exercise, so please, any help would be appreciated. [/QUOTE]
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What is the Corollary of the Nucleus and Image Theorem?
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