Definition. A function f from R^{n} to R^{m} is abstractly linear if it obeys the two rules
f(x_{1} + x_{2}) = f(x_{1}) + f(x_{2}) for all x_{1}, x_{2} in R^{n}.
f(c x) = c f(x) for all scalars c and all x in R^{n}.
Theorem. If f is an m \times n matrix, then f(x) = M x is always abstractly linear. Conversely, any abstractly linear function f from R^{n} to R^{m} is always given in this way by an m \times n matrix.
The definition of vector space may be found in the textbook.
The notion of vector space is an abstract concept. Statements about abstract vector spaces apply to all examples.
Three Examples
For each positive integer n R^{n} is a vector space.
The set of all differentiable functions on the interval a < x < b (in R^{1}) is a vector space when the sum of f and g is defined by (f + g)(x) = f(x) + g(x) and multiplication of a function f by a number c is defined by regarding c as a constant function, i.e., when c f is understood to mean the function given by (c f)(x) = c f(x).
The set of all functions f that satisfy the differential equation f'' = f is a vector space when addition of functions and the multiplication of a function by a scalar is understood as in the previous example.
Explain why a linear subspace of R^{n} is an example of a vector space.
Explain why a line through the origin of R^{n} is an example of a vector space.
Is the set of solutions in R^{3} of the equation x - 2 y + 5 z = 0 an example of a vector space?
Might the set of all m \times n matrices be a vector space?
Might the set of all n \times n invertible matrices be a vector space?