Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Linear with respect to each of two variables or positions. Used of functions or equations.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Linear (preserving linear combinations) in each variable.
 n. A bilinear function
from the GNU version of the Collaborative International Dictionary of English
 adj. Of, pertaining to, or included by, two lines.
from The Century Dictionary and Cyclopedia
 Consisting of or having reference to two lines: as, bilinear coördinates.
 Linear in two ways. Thus the form
Σ iΣ j aij xi yj is a bilinear form, that is, lineolinear.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 adj. linear with respect to each of two variables or positions
Etymologies
Examples

The far right is the 720 export from 1080i original from iMovie, and I found out that the closest effect was seen when I resized 1080i image to 720 using 'bilinear' interpolation, very cheap way to resize an image.

However, the polarization identities will allow you to recover the bilinear function from the quadratic one.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

EDIT: A note from The Unapologetic Mathematician that I should add: the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

Then in matrix form the metric looks like diag ( 1,1,1,1) as a bilinear form.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

They all use bilinear filtering rather than something reasonable like bicubic or lanczos or, really, anything that looks decent.

From the density matrix point of view, state vectors are a kluge that is done so that a quantum state can be converted from its natural bilinear condition, to an unnatural, but mathematically convenient linear one.

I realize this is “raw” data, but we also need to be aware that Hansen99 used 1950 as the hinge point for his bilinear adjustment.
A First Look at the USHCN Quality Classification « Climate Audit

We take advantage of the metadata accompanying the GHCN records, which includes classification of each station as rural (population less than 10,000), small town (10,000 to 50,000), and urban (more than 50,000), to calculate a bilinear adjustment for urban stations.

Also considered was a variety of nonlinear models, including the bilinear model which produced

Multiplier test for some new type of model, but not the bilinear model.
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