A linear function (in the sense of calculus) is one of the form $$f(x)=ax+b$$ where $a$ and $b$ are constants. The constant $a$ is the slope and the constant $b$ is the intercept of the function. This type of linear function occurs, for example, in linear regression.
In linear algebra, a linear function is a map $f$ between two vector spaces which is compatible with addition and scalar multiplication: $$f(x+y) = f(x) + f(y), \quad \quad f(ax) = a f(x),$$ where $x$ and $y$ denote elements of a vector space and $a$ denotes a scalar.
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