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In the following proof I am trying to show that

Prove that the function $T : R^3 → R^3$ defined by $T(w) = Proj_π(w)$ is a linear transformation.

My textbook has shown that it holds under addition; now I want to show it holds under scalar multiplication.

$$\begin{align}\operatorname{Proj}_\pi (cv) &= \frac{(cv)\cdot f_1}{f_1\cdot f_1} f_1 + \frac{(cv)\cdot f_2}{f_2\cdot f_2} f_2 \\ &= \frac{(cv)\cdot f_1}{f_1\cdot f_1} f_1 + \frac{c(v)\cdot f_2}{f_2\cdot f_2} f_2 \\ &= c\left[\frac{v\cdot f_1}{f_1\cdot f_1} \right] f_1 + c\left[\frac{v\cdot f_2}{f_2\cdot f_2} \right] f_2 \\ \end{align}$$ $$= cProj_\pi(v)$$

hardmath
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bjp409
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1 Answers1

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Just to give this question an answer, your proof looks fine. However, it could be made slightly better. Here's how I'd write it:

$\color{red}{\text{Let $v \in \Bbb R^3$ and $c\in \Bbb R$. Then}}$

$$\begin{align}\operatorname{Proj}_\pi (cv) &= \frac{(cv)\cdot f_1}{f_1\cdot f_1} f_1 + \frac{(cv)\cdot f_2}{f_2\cdot f_2} f_2 \\ &= \frac{\color{red}{c(v\cdot f_1)}}{f_1\cdot f_1} f_1 + \frac{\color{red}{c(v\cdot f_2)}}{f_2\cdot f_2} f_2 \\ &= c\left[\frac{v\cdot f_1}{f_1\cdot f_1} \right] f_1 + c\left[\frac{v\cdot f_2}{f_2\cdot f_2} \right] f_2 \\ &\color{red}{= c\left[\frac{v\cdot f_1}{f_1\cdot f_1}f_1 + \frac{v\cdot f_2}{f_2\cdot f_2}f_2\right]} \\ &= c\operatorname{Proj}_\pi(v) \end{align}$$

$\color{red}{\text{Thus $\operatorname{Proj}_\pi$ is homogeneous. This completes the proof the $\operatorname{Proj}_\pi$ is linear.}\ \ \ \square}$

Notice each of the parts I added or changed (in red). See if you can determine why I made these changes and how they improve the argument.