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The following is the mathematical definition of a Artificial Neuron,

$$\textbf{Activation function } f: \qquad y = f(w^t x) = f\left( \sum_{i=1}^n w_i x_i \right)$$

Given that $W$ is a vector, what does the symbol $t$ represent?

What operation does it represent on a vector $w$?

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  • $\begingroup$ $w^t x$ is the weighted average of $x.$ Were $w$ represents the weights. At least that is what it would be if all of the elements in $w$ are greater than $0$ and sum to $1.$ $\endgroup$ – Doug M Dec 21 '16 at 22:52
  • $\begingroup$ You used a lower-case $w$ and a capital $W$. If they both mean the same thing, you should use one or the other. $\endgroup$ – Michael Hardy Dec 21 '16 at 22:55
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The $t$ stands for transpose. The result of the matrix multiplication between $w^t$ and $x$ is then simply the scalar/dot product between $w$ and $x$, which in turn is just $\sum_{i=1}^n w_ix_i$:

$$w^tx = \begin{pmatrix} w_1\\ w_2\\ \vdots\\ w_n\end{pmatrix}^t\begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n\end{pmatrix} = \begin{pmatrix} w_1 \; w_2 \; \ldots \;w_n \end{pmatrix}\begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n\end{pmatrix} = w \cdot n = \sum_{i=1}^n w_ix_i$$

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  • $\begingroup$ Zillions of books say how matrix multiplication is defined, but nearly all of them omit to explain why it's defined that way, although that's easy to do. I wonder if some of that wouldn't help avert confusions about things like this. $\endgroup$ – Michael Hardy Dec 21 '16 at 22:57

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