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I was watching gilbert strang lectures on the projection matrix, at 5.17 he wrote that:

$a^T(b-xa) = 0 $

That means:

$xa^{T}a = a^Tb$

how can we write this expression?

In my opinion isn't it should be: $ a^Txa = a^Tb $ ?

Actually I am not good at linear algebra, so I know the question may be silly.

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    $\begingroup$ Can you define $x$, $b$, and $a$ in terms of their dimension? No one wants to watch your video. $\endgroup$
    – user762914
    Commented May 12, 2020 at 17:27

2 Answers 2

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Because $x\in \mathbb{R}$ and then $a^T.xa=xa^T.a=x\|a\|^2$

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Because, If you heard carefully,(at 2:50) he says that $\textbf{p}$ is scalar multiple of $\textbf{a}$ for some $x \in \mathbb{R}$

$$\textbf{p} = x.\textbf{a} , x \in \mathbb{R} , \textbf{a}^T. (x\textbf{a}) = x( \textbf{a}^{T}.\textbf{a})$$

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