Why vector space is named so?

Question

The thing I want to know is that why the term $``vector"$.

I had learned that vector is something that has magnitude and direction, and also the the elements of $\mathbb{R}^n$ i.e. $n$ tuples can be visualized as vectors . We can also perform vector addition and scalar multiplication over them. So this was fine. But again there are also vector spaces where I am not able to visualize the elements as the vectors like.

The set of $n\times n$ matrices whose elements are chosen from the set of real numbers forms a vector space over $\mathbb{R}$. Also the set of real valued continuous functions forms vector space.

So are this elements vectors or am I mistaken with the meaning of vectors?

Answer 1

The word vector originates from Latin, where it means "a carrier". It was first used in 18th century by astronomers, who were describing the motion of planets. For them, a vector was something that "carries" a point A to point B. It had a specific length and direction. So first vectors in mathematics/physics were vectors in the physical space.

Such vectors can be added, subtracted and multiplied by a number. In 19th century the term has been used to denote the elements of any set in which we have these operations appropiately defined. Such sets were named vector spaces.

Since you can add matrices of the same dimension to each other, and you can multiply them by numbers, they form a vector space, though a completely different one than the common space of vectors in the physical space.

Answer 2

The ordinary geometric vectors in $\mathbb R^2$ or in $\mathbb R^3$ are a special kind of a more general structure named vector space which obeys to the same rules and that we can extend to others object such as matrices, polynomials, etc. which can be added together and multiplied by scalar from any field (usually $\mathbb R$ or $\mathbb C$).