In linear algebra, the concepts of vector spaces, subspaces, and vectors are foundational. Here's an overview of each concept:
### Vector Spaces
A **vector space** (also called a linear space) is a collection of objects called **vectors**, which can be added together and multiplied by scalars. A vector space over a field \( F \) satisfies the following properties:
1. **Closure under Addition**: If \( \mathbf{u} \) and \( \mathbf{v} \) are