# Vector Space [ch.4]
Important Note: In chapter 10, we let F be a field.
## 4.1. Introduction to Vector Spaces
- Nothing here
## 4.2. Vector Spaces and Linear Transformations
- **Definition 4.1: Field**
- **Definition 4.3: *F* - vector space**
Note: Only the scalars are in F.
- **Proposition 4.4**
- **Definition 4.5: *F* - linear transformation**
## 4.3. Interesting Examples of Vector Spaces
## 4.4. Bases and Dimension
- **Natural Basis**
- **Definition 4.11: Finite Basis**
- **Definition 4.14: Span of $A$**
- **Proposition 4.15**
- **Theorem 4.16**
- **Theorem 4.18: Invariance of Dimension**
- **Definition 4.19: Dimension**
- **Lemma 4.23: Swap Lemma**
- **Lemma 4.24**

## Others
1. Vectors in Basis are linear independent.
2. Vectors in spanning set are not necessarily linear independent.
3. Vector space is a set of vectors.
4. Basis and spanning set can span the vector space, so Basis $\subseteq$ Spanning set.