1. Basic Vector Operations

1.1 Dot product (Inner product)

When we write a vector horizontally, there must be commas added between each two elements.

1.2 Norm

There are different definitions about Norm and this definition is called L2 Norm ## 1.3 Unit vector

2. Spanning Sets and Basis

2.1 Field

The addition and multipilication are not the same as ordinary ones. They can be defined as any kind, as long as the field axioms are satisfied, the field is legel. Namely, according to different definition, 2+3 is not necessarily 5. But here we just define a+b = a+b

2.2 Vector spaces

Under multipilication, only scalors are involved and no conditions about vector multiply vector are stated.

For example: \[If \quad V = \{\begin{pmatrix} a \\ b \end{pmatrix}| a, b \in R\} \quad is \quad a \quad vector \quad space\]

\[ Then \quad S = \{\begin{pmatrix} a \\ 0 \end{pmatrix}| a, b \in R\} \subset V \quad is \quad also \quad a \quad vector \quad space\]

There are two ways to check whether a set of vectors are linearly independent
1). Apporach one: Write the vectors in each colunm Here, non-trivial means that not all the solutions are 0 2). Apporach two: Write the vectors in each row

3. Geometry of Vector Spaces

When decomposing a vector of length n, we need to decompose it into n-1 orthogonal vectors and the final one, which is perpendicular to all the others, is the b - b1 - b2 - … - bn-1

Here all the parameters mean projections