Comparing Scalars Using Standard Comparison Operators

QUIZ #03 (30 points) NAME:____________________ (uppercase) 1. (20 points) Terms, Definitions, and (possible) Values A F 0(only) [+, -] B G +(positive) [+, 0, -] C Choices -(negative) D [-, 0] E [0, +] Assume that 𝐱 = [𝒙𝟏 , 𝒙𝟐 , ⋯ , 𝒙𝒏 ]𝑻 , where 𝒙𝒊 ∈ ℝ (i.e., 𝒙𝒊 is a scalar.); thus 𝐱 ∈ ℝ𝑛 (i.e., 𝐱 is an n-dimensional (column) vector.). For each of (1) – (12) in the table below, choose all that apply from the choices (A – T) in the first table. Term (01) (arithmetic) mean (02) median (03) mode (04) difference (05) distance (06) deviation (07) variance (08) standard deviation (09) skewness (10) kurtosis Definition (Provide a definition in a short sentence or in a formula.) Possible values (One from A – G) 2. (10 points) (Vector) Norms Similarly, WLOG, assume that the two column vectors (say, 𝐱 = [𝒙𝟏 , 𝒙𝟐 , ⋯ , 𝒙𝒏 ]𝑻 and 𝐲 = [𝒚𝟏 , 𝒚𝟐 , ⋯ , 𝒚𝒏 ]𝑻). We start by asking the following questions: (1) How to compare two scalars (say, 𝒙𝒊 and 𝒙𝒋 )? (2) How to get a difference between 𝒙𝒊 and 𝒙𝒋 (or between 𝒙𝒋 and 𝒙𝒊 )? (3) How to define a length from a difference between 𝒙𝒊 and 𝒙𝒋 (or between 𝒙𝒋 and 𝒙𝒊 )? (4) Can you the same questions and answers be directly applicable to x and y? (5) What are the properties of a vector? In other words, how do you define a vector? (6) How to calculate the direction of a vector? (7) How to calculate the magnitude of a vector? (8) What is the inner product of two vectors? (9) What is the outer product of two vectors? (10) What is the cross product of two vectors? (11) What is the dot product of two vectors? (12) What do you mean by being “orthogonal” between two vectors? (13) What is the norm of a vector? (14) What is the normalization of a vector? (15) What do you mean by being “orthonormal”? Term vector norm Definition p-norm 1. ‖x‖ ≥ 0 (non-negativity) and ‖x‖ = 0, iff 𝑥𝑖 = 0. (x is a zero vector.) 2. ‖αx‖ = |𝛼|‖x‖, for ∀𝛼 ∈ ℝ. (scaling) 3. ‖x + y‖ ≤ ‖x‖ + ‖y‖. (triangle inequality) ‖⋅‖: ℝ𝑛 → ℝ 𝑛 Properties (or related contents) 1 𝑝 ‖x‖𝑝 = (∑|𝑥𝑖 |𝑝 ) ℓ𝑝 -norm 𝑖=1 ℓ1-norm L1 norm Taxicab norm Manhattan norm ℓ2 -norm L2 norm Euclidean norm ℓ -norm supremum norm, Chebyshev norm, uniform norm, infinity norm, maximum norm p-norm normalization visualization of norms – – Relationship among norms – –

Collepals.com Plagiarism Free Papers

Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.

Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS

Why Hire Collepals.com writers to do your paper?

Quality- We are experienced and have access to ample research materials.

We write plagiarism Free Content

Confidential- We never share or sell your personal information to third parties.

Support-Chat with us today! We are always waiting to answer all your questions.