How do linear transformations preserve vector addition and scalar multiplication?

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modern algebra A simple algebraic curve, showing the graph of the algebraic equation y² = x³ + 1.

In linear algebra, a linear transformation is a function that maps one vector to another while maintaining the structural properties of the vector space. The term "linear" specifically refers to the transformation's ability to preserve the operations of vector addition and scalar multiplication.1

For a transformation $T$ to be considered linear, it must satisfy the following two conditions for any vectors $v$ and $w$ and any scalar $c$ :

Additivity: The transformation of the sum of two vectors is equal to the sum of their individual transformations:
$T (v + w) = T (v) + T (w)$
Homogeneity: The transformation of a scalar multiple of a vector is equal to the scalar multiplied by the transformation of that vector:
$T (c v) = c T (v)$

These properties ensure that the transformation respects the underlying linear structure of the space, allowing complex operations to be represented and computed efficiently using matrices.1