bilinjärfunktion

A bilinjärfunktion is a function of two variables that is linear in each variable when the other variable is held constant. In simpler terms, if you consider one variable and fix the other, the function behaves like a straight line. If you consider the other variable and fix the first one, it also behaves like a straight line.

Mathematically, a function $f(x, y)$ is bilinjär if for any constants $a$ and $b$, and for any variables $x_1, x_2, y_1, y_2$:

$f(ax_1 + bx_2, y) = af(x_1, y) + bf(x_2, y)$

and

$f(x, ay_1 + by_2) = af(x, y_1) + bf(x, y_2)$.

This property is known as linearity in each argument.

A common example of a bilinjärfunktion is multiplication: $f(x, y) = xy$. If we fix $y$, say to $c$, then $f(x, c) = cx$, which is a linear function of $x$. Similarly, if we fix $x$ to $d$, then $f(d, y) = dy$, which is a linear function of $y$.

Another example is the dot product of two vectors. If $u = (u_1, u_2)$ and $v = (v_1, v_2)$, their dot product is $u \cdot v = u_1v_1 + u_2v_2$. This function, when considered as a function of the components of the vectors, is bilinjär.

Bilinear functions are important in various areas of mathematics, including linear algebra, differential geometry, and the study of bilinear forms. They represent a generalization of linear functions to functions of multiple variables where linearity is preserved for each variable individually.