- #1

Portuga

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- 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)=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.

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.

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