Vector space
![Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.](/Images/godic/202502/19/Vector_add_scale.svg2303.png")
![A vector v in R2 (blue) expressed in terms of different bases: using the standard basis of R2 v = xe1 + ye2 (black), and using a different, non-orthogonal basis: v = f1 + f2 (red).](/Images/godic/202502/19/Vector_components_and_base_change.svg2303.png")
![Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces.](/Images/godic/202502/19/Vector_components.svg2303.png")
![A typical matrix](/Images/godic/202502/19/Matrix.svg2303.png")
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 in this context. 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 must satisfy certain requirements, called axioms, listed below.