Linear algebra uses linear functions to realistically approximate real-world functions. Furthermore, linear functions also reduce the complexity of a system.

Linear function definition

If x\bold{x} and y\bold{y} are any two objects(scalars, vectors, matrices, tensors) and α\alpha and β\beta are scalars, a linear function can be defined using two properties:

Additivity: f(x+y)=f(x)+f(y)f(\bold{x}+\bold{y}) = f(\bold{x})+f(\bold{y})

Homogeneity: f(αx)=αf(x)f(\alpha \bold{x}) = \alpha f(\bold{x})

We can combine these two properties in a single statement and say that a linear function must fulfill the following:

f(αx+βy)=αf(x)+βf(y)f(\alpha \bold{x}+\beta \bold{y}) = \alpha f(\bold{x})+\beta f(\bold{y})

Furthermore, we can extend these properties to more than two objects and scalars.

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