 #1
Portuga
 53
 6
 TL;DR Summary

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,
$$
\begin{align*}
& \dim\mathbb{R}^{2}=\dim\ker\left(F\right)+\dim\mathbb{R}\\
\Rightarrow & 2=\dim\ker\left(F\right)+1\\
\Rightarrow & \dim\ker\left(F\right)=21=1.
\end{align*}
$$
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.
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,
$$
\begin{align*}
& \dim\mathbb{R}^{2}=\dim\ker\left(F\right)+\dim\mathbb{R}\\
\Rightarrow & 2=\dim\ker\left(F\right)+1\\
\Rightarrow & \dim\ker\left(F\right)=21=1.
\end{align*}
$$
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.
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