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High School Algebra II: Help and Review26 chapters | 296 lessons

Instructor:
*Paul Bohan-Broderick*

Paul has been teaching many subjects in many different ways since he received his PhD in 2001.

Vector space models are representations built from vectors. These (relatively) simple models are especially good at representing phenomena that are not usually considered numerical and have been utilized in unexpected domains such as literary criticism.

Computers are tools for doing arithmetic-- adding, subtracting, multiplying and dividing integers. How can they also be powerful tools for learning about things that are not numerical?

One powerful answer is a **vector space model**. A **vector** is a number that has both a magnitude and a direction. Both magnitude and direction need to be measured with respect to the **space** in which the vector is defined. Each **dimension** of the space represents a feature of interest and a vector represents the extent to which the object of the model has those features. Thus, a vector is a list of numbers: one for each feature that is part of the model space. The *direction* of the vector is the one that from the origin of the space to the point defined by those numbers.

While at first this might sound very difficult and complex, you will see in the following sections that the vector space model can be applied to a wide variety of different contexts.

The easiest vector models to visualize come from physics. The most obvious vector model might describe the position of a particle in physical space using three numbers corresponding to measurements on three axes (length, depth and height). A more complicated physical vector model might include the three spatial dimensions and three dimensions corresponding to the speed in each of those directions. For example, a car might be one mile north and two mile east of my current location, but at the same elevation. The position of this car could be represented with the vector <1,2,0>. The first number represents position on the North-South axis, the second the position on the East-West access and the third represents positions on the up-down axis. If that car was travelling 60 mph due north, we could represent that as vector <1,2,0,60,0,0>, where the last three numbers represent speed in each of the three directions. In this example, if we had vectors representing different cars, we could compute their relative positions and velocities using trigonometry.

A text document can be represented as a vector. The vector space is defined by the terms that may be present in the text. For instance, each dimension in the space may represent a term in a dictionary of English. The dictionary defines the space. A text written in English would be a vector with a certain magnitude in each direction equal to the number of times that word or term appears.

If the word 'fish' appeared three times in a sentence, the vector would have a magnitude 3 in the *fish* direction. In a model space with the dimensions or features 'one', 'two', 'three', 'red', 'green', 'blue', and 'fish', the title of the Dr. Seuss classic *One Fish, Two Fish, Red Fish, Blue Fish* could be represented as <1,1,0,1,0,1,3>.

Many of Dr. Seuss' books are purposefully written with a vocabulary of only 50 words. Each of these books could be represented as a vector in 50-dimensional space or as a list of 50 numbers. Shakespeare's writing on the other hand contains around 28,829 specific words (or word types) and the vectors representing each play would be a very long lists of numbers.

Obviously, it is more rewarding to read a text rather than scan a list of numbers. However, representing a text in this way has many powerful applications, we will now look at two.

Representing texts as vectors allows an interested user to compute the similarity of two texts in the same way that the relative positions and velocities of two cars could be computed in the first example. Vector representations of texts make it relatively easy to measure the similarity of different texts by comparing the angles between the vectors that represent each text.

A program might find that *One Fish, Two Fish, Red Fish, Blue Fish* has some important similarities to *One, Two Buckle My Shoe* because they have similar, or close, vector representations.

Since authors tend to use the same words in different works, the output of a particular author will tend to be closer to each other in a vector space model then they are to the vectors representing texts written by other authors. A vector that falls far away from the vectors representing texts by the same author might have been improperly attributed. These techniques have been used as evidence when scholars are arguing about whether a certain play was written by Shakespeare or one of his contemporaries.

It is amazing how quickly and accurately the typical cell phone can offer predictions for the next word that a user might want to type. There are several ways that a text program might be making a prediction, but the simplest are based on vector space models. Whatever the user has inputted is represented as a vector. The computer searches through a set of vectors representing texts that have been typed in the past and finds the vectors that are closest to the input vector and suggest the words that are next on the most similar vectors.

Messages that contain the words 'I want ' would be much closer to vectors that continue with 'ice' (and then perhaps 'cream') or ' to' (and then maybe 'talk') than they would to vectors that continue with 'with' or 'not'. Both of these options would be poor grammar. The algorithm doesn't know that but it still wouldn't make these suggestions.

This seems like magic. There is no explicit representation of the meaning of the sentence, just word frequencies, but it often seems as if the text program knows or understands what you are going to type next. Often, the secret is just a vector space model. The representation of these words as vectors can transform problems about interpreting words into arithmetic problems.

Vectors are numbers represented in a space. This simple idea gives useful and interesting ways to describe physical space. However, a model doesn't have to describe physical space. If we interpret the dimensions of our model to measure other features, such as how often a certain word is used, we can construct numerical models of objects that don't seem to be numerical. These techniques have many powerful applications in everyday technology, such as the world prediction algorithms that are part of our texting applications.

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High School Algebra II: Help and Review26 chapters | 296 lessons

- What is the Correct Setup to Solve Math Problems?: Writing Arithmetic Expressions 5:50
- Understanding and Evaluating Math Formulas 7:08
- Expressing Relationships as Algebraic Expressions 5:12
- Evaluating Simple Algebraic Expressions 7:27
- Combining Like Terms in Algebraic Expressions 7:04
- Practice Simplifying Algebraic Expressions 8:27
- Negative Signs and Simplifying Algebraic Expressions 9:38
- Writing Equations with Inequalities: Open Sentences and True/False Statements 4:22
- Common Algebraic Equations: Linear, Quadratic, Polynomial, and More 7:28
- Defining, Translating, & Solving One-Step Equations 6:15
- Solving Equations Using the Addition Principle 5:20
- Solving Equations Using the Multiplication Principle 4:03
- Solving Equations Using Both Addition and Multiplication Principles 6:21
- Collecting Like Terms On One Side of an Equation 6:28
- Solving Equations Containing Parentheses 6:50
- Translating Words to Algebraic Expressions 6:31
- How to Solve One-Step Algebra Equations in Word Problems 5:05
- How to Solve Equations with Multiple Steps 5:44
- How to Solve Multi-Step Algebra Equations in Word Problems 6:16
- Distributive Property: Definition, Use & Examples 6:20
- Vector Space Model: Examples
- Substitution Property of Equality: Definition & Examples 2:57
- Translational Symmetry: Definition & Examples 3:26
- Go to Algebra II - Algebraic Expressions: Help & Review

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