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