Similar to Tom's answer,
a vector is a mathematical quantity with both a magnitude and a direction.
Useful for distinguishing between speed and velocity but little else. The above is a typical definition from a physics textbook I had on the shelf; here in British Columbia, vectors are introduced in high school physics but not high school math. By the time students get to linear algebra in first- or second-year university, it can be hard to convince them that a real number (much less a polynomial) can be a vector. Usually, you have to resort to "a real number does too have a direction: positive or negative" and even then they don't believe you because
a scalar is a mathematical quantity with a magnitude and no direction
and so if real numbers are vectors, how can they be scalars?
Don't even ask about function spaces.