Definition Of Vector Space Math

A vector space also called a linear space is a collection of objects called vectors which may be added together and multiplied scaled by numbers called scalars scalars are often taken to be real numbers but there are also vector spaces with scalar multiplication by complex numbers rational numbers or generally any field the operations of vector addition and scalar multiplication.

Definition of vector space math. The addition is just addition of functions. In mathematics and physics a vector is an element of a vector space. F1 f2 n f1 n f2 n. Scalar multiplication is just as simple.

To see more detailed explanation of a vector space click here. Historically vectors were introduced in geometry and physics typically in mechanics before the formalization of the concept of vector space therefore one often talks about vectors without specifying the vector space to. If u and v are any vectors in v then the sum u v belongs to v. This way the theorems start with the phrase let v be a vector space instead of the vague rambling phrase above.

Vector space is defined as a set of vectors that is closed under two algebraic operations called vector addition and scalar multiplication and satisfies several axioms. Now when we recall what a vector space is we are ready to explain some terms connected to vector spaces. For many specific vector spaces the vectors have received specific names which are listed below. But mathematicians like to be concise so they invented the term vector space to mean any type of mathematical object that can be multiplied by numbers and added together.

A vector space is a set v on which two operations and are defined called vector addition and scalar multiplication. N ℜ here the vector space is the set of functions that take in a natural number n and return a real number. C f n cf n. In the above example the notion of adding and multiplying 2 polynomials was already laid.

This works well when we are shown that in a sense polynomial functions are vector spaces however i m left with a slight confusion. Denition of a vector space. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers complex numbers etc.

A vector space or a linear space is a group of objects called vectors added collectively and multiplied scaled by numbers called scalars. The way a vector space is introduced is by saying that vector space is a set v on which two operations and are defined called vector addition and scalar multiplication.