linear transformation
A linear transformation is a function that takes vectors to vectors while respecting two rules: it must spread out across addition and scaling, so T(a*u + b*v) = a*T(u) + b*T(v). In short, it does not matter whether you combine first and then transform, or transform first and then combine.
Picture a grid drawn on the plane. A linear transformation keeps the grid lines straight and evenly spaced, and it leaves the origin fixed. Lines stay lines, parallel stays parallel. Familiar examples are rotations, scalings, shears, and projections onto a line.
What it is NOT: anything that bends lines, or that shifts the whole picture sideways (a plain move that does not fix the origin is not linear). That clean, grid-preserving behavior is exactly what makes these maps so easy to compute with.
A horizontal stretch by 2. Check it is linear: T(a*u + b*v) really does equal a*T(u) + b*T(v).
A common surprise: shifting everything by a fixed amount (called a translation) is NOT a linear transformation, because it moves the origin. Such maps are called affine instead.