I'm afraid my maths is so feeble I was not sure how to entitle the question. Essentially I was hoping to ask what the below passage means in the context of the passage that follows? (If that is reasonable way to ask questions on maths)
From Machine Learning: The Art and Science of Algorithms that Make Sense of Data by Peter Flach, page 4:
"There are a number of useful ways in which we can express the SpamAssassin classifier in mathematical notation. If we denote the result of the $i$-th test for a given e-mail as $x_i$, where $x_i = 1$ if the test succeeds and $0$ otherwise, and we denote the weight of the $i$-th test as $w_i$, then the total score of an e-mail can be expressed as $\sum_{i=1}^n w_i x_i$, making use of the fact that $w_i$ contributes to the sum only if $x_i = 1$, i.e., if the test succeeds for the e-mail. Using $t$ for the threshold above which an e-mail is classified as spam (5 in our example), the ‘decision rule’ can be written as $\sum_{i=1}^n w_i x_i > t$.
"Notice that the left-hand side of this inequality is linear in the $x_i$ variables, which essentially means that increasing one of the $x_i$ by a certain amount, say $\delta$, will change the sum by an amount $(w_iδ)$ that is independent of the value of $x_i$. This wouldn’t be true if $x_i$ appeared squared in the sum, or with any exponent other than $1$."
Recall that the equation of a line is $y=mx+b$, where $m$ and $b$ are constants. It seems that when they say "linear in the $x_i$," what they mean is that "if you pick one of the $x_i$, say $x_7$, and you treat every other variable in your equation as a constant, the resulting equation you get is the equation of a line."
They may actually means something slightly stronger, which I can't quite tell from context. Namely, they may also have the requirement that if all the $x_i$ are equal to zero, then the overall expression is zero too. This holds true in the expression $\sum x_i w_i$, but there is not enough context to say that it holds true everywhere in the document you're using.
If they do mean linear in this stronger sense, then it fits with the definition of linear used in linear algebra. Namely, let $x=(x_1,x_2,\ldots, x_n)$ be the vector whose components are your $x_i$. Let $w$ be a vector representing all the other variables in your expression, which we will write as $f_w(x)$ because, while the expression depends on $w$, we are going to want to fix a value of $w$ and just look at how things depend on $x$. So treat $w$ as a constant, so that $f_w(x)$ actually is just a function of $x$. To say that it is linear is to say that $f_w(ax)=af_w(x)$ and $f_w(x+y)=f_w(x)+f_w(y)$.