I'm struggling with understanding the underlying vector multiplication that are used to define a decision boundary for a binary classification. According to my textbook, this line may be expressed as: $y(x) = w^T.x + w_0=0$
Here, $w$ represents the parameter vector, and $x$ represents the input vector. I'm confused about the dimensions of these $2$ vectors in a situation where for example, 'height' and 'weight' are used to class somebody as 'overweight' or 'underweight'. We now have $2$ input variables, and let's assume we have $5$ rows of data.
Would this mean that $x$ is a ($5\times 2$) vector, and that $w^T$ is a ($2\times ?$) vector? If so, how would the dot product translate into a line? Because we would end up with a ($5\times ?$) vector, not a scalar.
I understand how $y(x)$ is used to classify a point, because here that would be a ($1\times 2$) . ($2\times 1$) dot product, giving a scalar. But in terms of $y(x)=0$, I'm not able to wrap my head around it.
I apologise if this is something really basic or if I'm just being silly and missing something glaring, but I really do appreciate any guidance!