# Linear algebra/Introductory definitions

### Vector spaces, vector operations, matrix operations

A vector space is comprised of a "scalar" field (for example, the real numbers), a set of "vectors", and two binary operations which must satisfy certain properties. One operation is vector addition, a binary operation that takes two vectors yields another vector. The other is called scalar multiplication, a binary operation that takes a scalar and a vector and yields a vector. The study focuses primarily on matrices, which can be thought of as two-dimensional arrays of elements from the scalar field of our vector space.

Most explorations focus on real and complex vector spaces. For instance, the set of vectors ${\displaystyle \{[x,y,z]:x,y,z\in \mathbb {R} \}}$ is called ${\displaystyle \mathbb {R} ^{3}}$. When we want to speak more generally, we will assume an arbitrary dimensionality and use the notation like ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle \mathbb {C} ^{n}}$. Before going further, the reader should become familiar with a few basic operations on these objects. The following videos demonstrate these operations on real-valued matrices and vectors.