###### tags: `Linear Algebra` `LA01` # L04 Norm and Distance --- ## From Last week - Inner Product of Vectors - Application of Inner product, Net Present Value - Sentiment Analysis - How to wrangling with text data in Python --- ## This week - Norm of vector - Distance of vectors - Triangle Innequality - Knn intro --- ### 1. Norm A Euclidean Norm of a vector is defined as: $$||x|| = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2} = \sqrt{x^Tx}$$ he right part of the equation is in the form of inner product. Recalls that : $$a^T b = a_1b_1 + a_2b_2 + \dots + a_nb_n$$ ---- ---- ### 1.1 Actually, **Norm** is a more formal and abstruct concept, In our module, we only consider **Euclidean Norm**, which is simply, the **Length** of the vector. :bulb: *Quick Quiz*: What is the norm of the vector $\begin{bmatrix} 1\\1\end{bmatrix}$ ? --- ### 2. Properties of norm Some Important properties of the Euclidean norm are given below, given x adn y are vectors with same dimensions, $\beta$ is a scalar. - Euclidean Norm is always non-negative, and $\begin{Vmatrix}\beta x\end{Vmatrix}=|\beta|\begin{Vmatrix}x\end{Vmatrix}$ - Triangle inequality. $\begin{Vmatrix}x + y\end{Vmatrix}\le\begin{Vmatrix}x\end{Vmatrix} + \begin{Vmatrix}y\end{Vmatrix}$ - Definiteness. $\begin{Vmatrix}x\end{Vmatrix} = 0$ only if $x = 0$ --- ### 3. Distance With help from the Norm(Length), it will be easy for us to define "Distance of two vectors": $$dist(a, b) = \begin{Vmatrix}a - b\end{Vmatrix}$$ ---  --- ### 3.1 Triangle Inequality Triangle Inequality in the form of distance, let - $dist(a,b) = \begin{Vmatrix}a - b\end{Vmatrix}$, - $dist(b,c) = \begin{Vmatrix}b - c\end{Vmatrix}$, - $dist(a,c) = \begin{Vmatrix}a - c\end{Vmatrix}$, then, $$\begin{Vmatrix}a - c\end{Vmatrix} \le \begin{Vmatrix}a - b\end{Vmatrix} + \begin{Vmatrix}b - c\end{Vmatrix}$$ ---  --- ### Nearest Neighbor ---  --- ### Thank you! :sheep: Python Time!