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 $$ \{[x,y,z] : x,y,z \in \mathbb{R}\}$$ is called $$ \mathbb{R}^3 $$. When we want to speak more generally, we will assume an arbitrary dimensionality and use the notation like $$ \mathbb{R}^n $$ and $$ \mathbb{C}^n$$. 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.

Primer on basic matrix operations

 * 1) Youtube: Element-wise Matrix Operations
 * 2) Youtube: Matrix multiplication
 * 3) Youtube: Gaussian Elimination