tuple in Linear algebra are called vector.
A vector is a list of scalar (real number) used to represent a function
When the letters are in bold in a formula, it signifies that they're vectors,
To represent the below function:
<math> \begin{array}{rrr} 0 & \mapsto & 8 \\ 1 & \mapsto & 7 \\ 2 & \mapsto & −1 \\ 3 & \mapsto & 2 \\ \end{array} </math>
where:
we use the following dictionary 4-vector:
<math>\{0:8, 1:7,2:−1, 3:2\}</math>
that can be simplified as this list of number:
<math>[8, 7,−1,2]</math>
Technically, a vector is:
Python’s dictionaries can represent such vectors, e.g.
{0:8, 1:7, 2:-1, 3:2}
The following convention is often adopted: entries with value zero may be omitted from the dictionary.
A class Vec with two instance variables (fields):
A list L must be viewed as a function where the domain is the index of the value ie {0, 1, 2, . . . , len(L)}.
Example: <math>[8,7,-1,2]</math>
<math>[8, 7,−1,2]</math>
<math>[8, 7,−1]</math>
Used to represent
In natural language processing, a document is represented by a bag of words model by a function <math>f : WORDS \mapsto \mathbb{R}</math> specifying, for each word, how many times it appears in the document.
For any single document, most words in the word dictionary are of course not represented. They should be mapped to zero but a convenient convention for representing vectors by dictionaries allow to omit pairs when the value is zero.
Example representing a WORDS-vector over : “The rain in Spain falls mainly on the plain” would be represented by the dictionary
{’on’: 1, ’Spain’: 1, ’in’: 1, ’plain’: 1, ’the’: 2, ’mainly’: 1, ’rain’: 1, ’falls’: 1}
(for cryptography/information theory)
Mathematics - Probability distribution function e.g. {1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6}
{(0,0): 0, (0,1): 0, (0,2): 0, (0,3): 0,
(1,0): 32, (1,1): 32, (1,2): 32, (1,3): 32,
(2,0): 64, (2,1): 64, (2,2): 64, (2,3): 64,
(3,0): 96, (3,1): 96, (3,2): 96, (3,3): 96,
(4,0): 128, (4,1): 128, (4,2): 128, (4,3): 128,
(5,0): 160, (5,1): 160, (5,2): 160, (5,3): 160,
(6,0): 192, (6,1): 192, (6,2): 192, (6,3): 192,
(7,0): 224, (7,1): 224, (7,2): 224, (7,3): 224 }
The D-vector whose entries are all zero is the zero vector, written or just 0.
To test if a vector v should be considered a zero vector, you can see if the square of its norm is very small, e.g. less than <math>10^{-20}</math>
A vector most of whose values are zero is called a sparse vector.
If no more than k of the entries are non-zero, we say the vector is k-sparse. A k-sparse vector can be represented using space proportional to k. For instance, when we represent a corpus of documents by WORD-vectors, the storage required is proportional to the total number of words in all documents.
Most signals acquired via physical sensors (images, sound, …) are not exactly sparse.
Vectors that are mutually orthogonal and have norm 1 are orthonormal
Set of all 4-vectors over is written . See Linear Algebra - Function (Set)
Example of gf2 Set: is the set of 5-element bit sequences, e.g. [0,0,0,0,0], [0,0,0,0,1], …
n-vectors over can be visualized as arrows in
Mathematicians | ||
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William Rowan Hamilton | William Rowan Hamilton, the inventor of the theory of quaternions. The quaternions are a number system that extends the complex numbers. | |
Josiah Willard Gibbs | Developed vector analysis as an alternative to quaternions. |