Determine Vector Space

The operation vector addition must satisfy the following conditions.

Determine vector space. E a bv ab v. For example the denition from the course notes has two additional axioms. Dierent authors write the denition of a vector space dierently. In the questions below i should determine whether each of the sets given is a vector space or not.

The basic example is dimensional euclidean space where every element is represented by a list of real numbers scalars are real numbers addition is componentwise and scalar multiplication is multiplication on each term separately. The vector space of lines through the origin of r2. A at v a m3 3. The vector space of polynomials of arbitrary degree.

The vector space of m x n matrices. From our notes i know that it must satisfy the conditions to be a vector space. Vector space a vector space is a set that is closed under finite vector addition and scalar multiplication. Prove the following vector space properties using the axioms of a vector space.

The cancellation law the zero vector is unique the additive inverse is unique etc. The sum of two vectors must be a vector and the multiple of a vector by a scalar is a vector. If u and v are any vectors in v then the sum u v belongs to v. But i have no idea how to apply any of these conditions to the question at all.

The vector space of n component vectors. To check that is a vector space use the properties of addition of functions and scalar multiplication of functions as in the previous example. G a bu a u b u. 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.

F 1 v v. 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. A v w w v. For all vectors u and v in v u v v u.

X y 2z 3 v p p4x. V x y z r3. C v 0 0 v v. The zero vector space.

D v v 0. B u v w u v w. We have looked at a variety of different vector spaces so far including. P0 p1 0 v a m3 3.

Determining whether a set is a vector space.