In Fisher's LDA, what is $a_i = w^T x_i$ if $w$ is an unit vector and $x_i$ is an observation?

My notes write that:

projection lengths of data into it (unit vector $w$) as $a_i = w^T x_i$

So is $a_i$ some length or what?

Or perhaps it's a reduction of $x_i$ into lower dimension?


gives on p. 5 that $w^T x_i$ would be a scalar, which is "the distance of projection of $x_i$ from the origin.

But I have difficulties in recalling, what is this in standard vector algebra terms?


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