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Scalars and Vectors

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(Duke University - Cheng-Yu Chen)

 

- Vectors and Scalars

In linear algebra, scalars are real numbers that relate to vectors through scalar multiplication. Scalar multiplication is the process of multiplying a vector by a number to produce another vector. The term "scalar" comes from the idea that a scalar scales vectors.

Vectors and scalars are two types of quantities used in physics and math. Scalars are quantities that only have magnitude, or size. Vectors have both magnitude and direction. 

Here are some examples of scalars and vectors: 

  • Scalars: Speed, distance, mass, density, pressure, volume.
  • Vectors: Velocity, displacement, acceleration, force, and mass

Here are some other examples of vectors:
  • 60 miles per hour north
  • 50 meters north

To remember the difference between vectors and scalars, you can ask yourself if there is a direction. For example, you would say "50 meters north" for displacement, but "50 meters" for scalars.
 

Please refer to the following for more details:

 

- Vector Spaces

So if you want to do any linear algebra, vector spaces are one of the foundations to understanding everything.  

A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (“scaled”) by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces. The methods of vector addition and scalar multiplication must satisfy specific requirements such as axioms.  

Here are some examples of vector spaces: 
  • Arrows in the plane
  • Ordered pairs of numbers
  • Coordinate space
  • Complex numbers and other field extensions
  • Function spaces
  • Linear equations
  • The set {0,1} with arithmetic modulo 2
  • The vector space of polynomials in x with degree at most 2 over the real numbers
  • The set of magic matrices
  • The solution set to any homogeneous linear equation

The simplest example of a vector space is {0}, which only contains the zero vector.
 
 

- Characteristics of Vector Spaces

In mathematics and physics, a vector space is a set of elements that can be added together and multiplied by numbers called scalars. Vector spaces are also known as linear spaces. 

Here are some characteristics of vector spaces:

  • The elements of a vector space are sets of numbers.
  • Each element is a list of objects with a specific length, called vectors.
  • The elements of a vector space are usually referred to as n-tuples, where n is the length of each element.
  • Scalars are often real numbers, but can also be complex numbers or elements of any field.
  • Vectors can be added together and multiplied by scalars while preserving arithmetic properties like associativity, commutativity, and distributivity.
  • Vector spaces are mathematical objects that capture the geometry and algebra of linear equations.
  • Vectors are used to represent many things, including forces like gravity, acceleration, friction, and stress and strain on structures.
 

- Properties of Vector Spaces

There are some properties to vector spaces: V1, V2, and V3. Big V is the set of all vectors. Little v is just a vector. A set V of vectors is called a vector space if it satisfies properties V1, V2, and V3. 

Here are some properties of vector spaces:

  • Addition: The sum of any finite list of vectors can be calculated in any order, and the solution will be the same.
  • Negation: The negation of 0 is 0. The negation of any negative value of a vector is the vector itself.
  • Additive identity: Every vector space has a unique additive identity.
  • Linear span: The span of a set of vectors is the linear space formed by all the vectors that can be written as linear combinations of the vectors in the set.
  • Basis: A basis of a vector space is a set of vectors that are linearly independent and span the vector space.
 

- Axioms of Vector Spaces

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. 

Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms.

 
Here are some axioms of vector spaces: 
  • u+v is in V.
  • u+v=v+u.
  • u+(v+w)=(u+v)+w.
  • V has a zero vector 0 such that for every u∈V, u+0=u.
  • For every u∈V, there is a vector in V denoted by −u such that u+(−u)=0.
  • cu is in V.
  • c(u+v)=cu+cv.
  • (c+d)u=cu+du.

A vector space is made up of three objects: a set and two operations.
 
 
 

[More to come ...]

 

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