I find the second point interesting in the context of functions. Can it be modified to have an "iff" instead of "if". If it becomes a hard requirement, we might be able to say that all functions that form vector spaces will always have a basis for the space. It would also be interesting to note that there will be the number of elements in the basis might not be finite.
keenan
@dranzer In finite dimensions, linear functions and functions obtained by taking a linear combination of a fixed set of vectors are indeed the same. Maybe consider that (i) every finite dimensional linear function can be represented by a matrix, and (ii) matrix-vector multiplication is just taking a linear combination of the matrix columns, using the vector entries as coefficients.
I find the second point interesting in the context of functions. Can it be modified to have an "iff" instead of "if". If it becomes a hard requirement, we might be able to say that all functions that form vector spaces will always have a basis for the space. It would also be interesting to note that there will be the number of elements in the basis might not be finite.
@dranzer In finite dimensions, linear functions and functions obtained by taking a linear combination of a fixed set of vectors are indeed the same. Maybe consider that (i) every finite dimensional linear function can be represented by a matrix, and (ii) matrix-vector multiplication is just taking a linear combination of the matrix columns, using the vector entries as coefficients.