Elements $\mathbf{v}\in {\mathbb{R}}^n$ are referred to as vectors

$$\mathbf{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]$$

where $v_i$ is a real number and ${\mathbf{v}}^{⊺}=(v_1,v_2,\dots ,v_n)$ , the ${}^{⊺}$ denotes the transpose operator. For two vectors $\mathbf{a}=(a_1,a_2,\dots ,a_n{)}^{⊺}\in {\mathbb{R}}^n$ and $\mathbf{b}=(b_1,b_2,\dots ,b_n{)}^{⊺}\in {\mathbb{R}}^n$ , we define the following properties:

• addition: $\mathbf{a}+\mathbf{b}=(a_1+b_1,a_2+b_2,\dots ,a_n+b_n{)}^{⊺}\in {\mathbb{R}}^n$
• multiplication by scalar: $\alpha \mathbf{a}=(\alpha a_1,\alpha a_2,\dots ,\alpha a_n{)}^{⊺}\in {\mathbb{R}}^n$
• scalar product: ${\mathbf{a}}^{⊺}\mathbf{b}={\sum }_{i=1}^n{a}_i{b}_i\in {\mathbb{R}}^n$

A linear subspace $L\subseteq {\mathbb{R}}^n$ is a set that holds:

• for every $\mathbf{a},\mathbf{b}\in L$ it holds that $\mathbf{a}+\mathbf{b}\in L$
• for every $\alpha \in \mathbb{R},\mathbf{a}\in L$ it holds that $\alpha \mathbf{a}\in L$

An affine subspace $A\subseteq {\mathbb{R}}^n$ is a set that is represented as:

• $\mathbf{v}+L=\{\mathbf{v}+\mathbf{x}|\mathbf{x}\in L\}$ for some vector $\mathbf{v}\in {\mathbb{R}}^n$ and linear subspace $L\subseteq {\mathbb{R}}^n$

Rank

The rank of a matrix $A$ is the dimension spanned by its columns. Thus, the maximal number of linearly independent columns in $A$, which in turn, is equal to the dimension spanned by its rows. The column rank and the row rank are always equal.

Trace

The trace ^[1] of a square matrix $A$, is the sum of the elements of the main diagonal, e.g. $I^3$ where $I$ is the identity matrix with three dimension has the trace of $3$.

$$\text{tr}(I^3)=3$$

References