PolarSPARC |
Introduction to Linear Algebra - Part 1
Bhaskar S | 01/29/2022 |
Basic Definitions
The following are some of the basic definitions that will be referred in to following paragraphs:
Name | Description |
---|---|
Scalar | is just a number (or a constant) and is represented using the lowercase greek alphabets -
|
Vector | a quantity that has both magnitude and direction, and is represented as a lowercase letter
with an arrow on the top - |
Matrix (plural Matrices) | a rectangular table of mathematical elements that is arranged in rows and columns, and is
represented with an appercase letter. Example: |
Linear Algebra is the branch of mathematics that deals with Vectors and Matrices.
Vectors
The following are some of the characteristics of Vectors:
is an ordered list of numbers (integer, real, rational, irrational, etc.), that are represented inside square brackets
[] or parentheses (), and can be column-oriented
the number of elements in the vector is called the Dimensionality of the vector
to perform any mathematical operations on two (or more) vectors, they must have the same dimensionality
to perform addition of two vectors, just add the corresponding elements of the vector. Example:
to perform subtraction of two vectors, just subtract the corresponding elements of the vector. Example:
to perform multiplication of a vector with a scalar, just multiple the scalar with each of the elements in the vector.
Example:
the transpose of a column vector
Example:
if
Dot Product of Vectors
The Dot (also known as the Scalar or
Inner) product of two vectors (of same dimensionality) is a single scalar. To perform the dot product of the two
vectors, multiply the corresponding elements of the two vectors and then sum each of the individual products. In other words,
if
Let us look at an example for the dot product.
Example-1 | Find the dot product of the vectors |
---|---|
The dot product is computed using the formula That is, |
The following are some of the properties of Dot Product (of vectors):
Associative :: true with scalar -
Commutative ::
Distributive ::
Magnitude of a Vector
The Magnitude (also known as the Length or
Norm) of a vector
We will use a geometric perspective to find the magnitude of a vector for a better intuition. A vector
The following graph is the illustration of the vector:
Using the Pythagorean Theorem, we can infer the length of
Extending this concept to a vector
Geometric Perspective of the Dot Product
From a geometric perspective, the dot product of two vectors
In other words,
Rearranging the equation, we get:
Or,
The following graph is the illustration of the two vectors:
From the triangle above, using the Pythagorean Theorem, we can infer the following:
Using Trigonometry, we can infer the following:
Expanding the equation (2) from above,
That is,
Replacing the first term with
Replacing q from equation (4) above, we get
Simplifying, we get
This is the equation for the Law of Cosines.
NOTICE that when
From geometry, we can infer
In other words,
We know
Substituting in the equation for the law of cosines, we get:
Simplifying the equation, we get:
For orthogonal vectors
Outer Product of Vectors
The Outer product of two vectors (of same or different dimensionality) results in a matrix.
In other words, if
An example will make it clear
Let
Let
For each element
Unit Vector
A Unit Vector is a vector whose magnitude (or length) equals one (1), that is,
Given a vector
Given
Therefore, a Unit Vector for
Let us look at an example now.
Example-2 | Find the Unit Vector for the vector |
---|---|
We know a Unit Vector of a given vector We also know RESULT: the Unit Vector for |
Direction of a Vector
The Direction is the second aspect of a vector
We will use a geometric perspective to find the direction of a vector for a better intuition. A vector
The following graph is the illustration of the vector:
From Trigonometry, we know
That is,
Similarly,
Therefore,
Extending this concept to a vector
Field
As indicated above, a Dimension is the number of elements in a vector. To get an intuitive understanding, one can look at it from a geometric perspective - a one-dimensional vector represents a line, a 2-dimensional vector represents a plane, a 3-dimensional vector represents a 3-dimensional space (also known as the Euclidean Space), and so on.
A Field is a set of numbers in a particular Dimension, on which, the four basic mathematical
operations, namely, addition, subtraction, multiplication, and division are valid. In mathematics, a Field is reprseneted as
a hollow capital letter, such as
One can associate Dimensionality with a Field by raising the Field letter with the Dimension. For example, the Field
Subspaces
A Subspace is the set of all vectors that can be created by all possible linear combinations
of vector-scalar multiplication for a given set of vectors in
In other words, a vector subspace must satisfy the following criteria:
Closure Property - must be closed under addition and scalar multiplication
Origin Property - must contain the Zero vector (a vector with all zero elements, which is equivalent to the origin in a 2-dimensional case)
The simplest linear combination is a scalar multiplication operation (scaling) of a vector. That is, for a given vector
In mathematical terms, if
Linear Dependence
A set of vectors are said to be Linearly Dependent if some scalar combination of the vectors can form the zero vector (a vector with all zero elements).
Mathematically,
The following set of vectors from the 3-dimensional space are linearly dependant:
We can show
Basis Vector
A Basis is a set of vectors that are linearly independent of each other and can be linearly combined to generate other vectors in the vector space they belong to.
The unit vectors