4- Matrix Multiplication

A neuron with multiple inputs can calculate its output using a dot product:

$$y=w\cdot x+b$$

When we have many inputs and many neurons, we need to calculate many dot products at once.

Matrix multiplication lets us do this.

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4.4 Vectors as Matrices

A weight vector can be written as:

$$w= \begin{bmatrix} w_1\\ w_2\\ w_3 \end{bmatrix}$$

and the input vector:

$$x= \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$$

To calculate the dot product, we use the transpose of (w):

$$w^Tx = \begin{bmatrix} w_1&w_2&w_3 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$$

Expanding:

$$\boxed{w^Tx=w_1x_1+w_2x_2+w_3x_3}$$

Therefore:

$$\boxed{y=w^Tx+b}$$

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