Vector

Characteristics

• Arrow in a hyperplane
• Magnitude (size)
• Direction

$\vec{v}=[v_1{v}_2{v}_3...v_n]$

$\vec{u}=[u_1{u}_2{u}_3...u_n]$

Norm

The vector norm is a way to measure it’s magnitude.

L1-Norm

Also known as Manhattan distance or Taxicab norm.

$\Vert \vec{v}{\Vert }_1={\sum }_{i=1}^n|v_i|$

L2-Norm

Is the most popular norm, also known as the Euclidean distance (norm). It is the shortest distance to go from one point to another.

$\Vert \vec{v}{\Vert }_2=\sqrt{{\sum }_{i=1}^n{v}_i^2}$

Vector and Matrix Operations

Vector Sum

$$\vec{v}+\vec{u}=\left[\begin{array}{cccc}{v}_1+u_1& v_2+u_2& \cdots & v_n+u_n\end{array}\right]$$

Vector Difference

$$\vec{v}-\vec{u}=\left[\begin{array}{cccc}{v}_1-u_1& v_2-u_2& \cdots & v_n-u_n\end{array}\right]$$

Multiply Vector by a Scalar

$$\lambda \vec{v}=\left[\begin{array}{cccc}\lambda v_1& \lambda v_2& \cdots & \lambda v_n\end{array}\right]$$

Multiply Vector vs Matrix

The number of matrix’s columns $(3)$ must be equal to the vector row’s number $(3)$.

$${\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right]}_{3\times 3}.{\left[\begin{array}{c}a\\ b\\ c\end{array}\right]}_{3\times 1}={\left[\begin{array}{c}10\\ 15\\ 12\end{array}\right]}_{3\times 1}$$

Vector and Matrix Transpose

The transpose is an operation that flips a matrix or vector over its main diagonal, switching its rows and columns. The transpose converts an $m\times n$ matrix into an $n\times m$ matrix.

For a vector $\vec{v}$ and a matrix $\mathbf{A}$:

$$\vec{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]\mathbf{A}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right]$$

The respective transposes are:

$${\vec{v}}^{⊺}=\left[\begin{array}{cccc}{v}_1& v_2& \cdots & v_n\end{array}\right]{\mathbf{A}}^{⊺}=\left[\begin{array}{cccc}{a}_{11}& a_{21}& \cdots & a_{m1}\\ a_{12}& a_{22}& \cdots & a_{m2}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \cdots & a_{mn}\end{array}\right]$$

Dot Product

Also known as inner product, this operation gives an intuition about the projections between the two vectors.

Given the two vectors:

$$\mathbf{a}=\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right]\text{and}\mathbf{b}=\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]$$

The dot product is given by:

$$\mathbf{a}\cdot \mathbf{b}=\sum _{i=1}^n{a}_i{b}_i=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$$

The dot product also can be structured as a matrix multiplication:

$$\mathbf{a}\cdot \mathbf{b}={\mathbf{a}}^{⊺}\mathbf{b}=\left[\begin{array}{cccc}{a}_1& a_2& \cdots & a_n\end{array}\right]\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$$

• $\mathbf{a}\cdot \mathbf{b}>0$ : Positive projection
• $\mathbf{a}\cdot \mathbf{b}=0$ : Ortogonal vectors
• $\mathbf{a}\cdot \mathbf{b}<0$ : Negative projection