Linear Algebra I :Outline of Vector Space
Wisdom is not a product of schooling but a lifelong attempt to acquire it — Albert Einstein
First of all thanks for supporting and those valuable comments❤ Thanks to your feedback I realized this type of concept wise discussion is more helpful for y’all and so I have decided along with my regular content I’ll try to bring this type of 5min concept summaries too. This will also give you the insights and we can discuss how to approach a problem.
Let’s warm up our senses with some definitions which are required for us to move forward.
Binary Composition
Let’s say we are considering a non-empty set V. Then Binary Composition is nothing but a operation function from V*V into V allowing composition of two elements to form a third element.
And the collection of these operations in the set S in finite arity and with a finite set of identities is called a Algebraic Structure.
Internal Composition
If V is any set, then the composition * is said to be internal composition on V if
then,
and it is unique. Internal composition is also a Binary Composition.
External Composition
An operator •(blob) is said to be external composition on a set V over another set F if for all a∈ F and v ∈V , a •v ∈V and it is unique.
Field
A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field).
[This is an advanced compared to our zone of discussion right now. If we start analyzing it’s structure and meaning of each word in definition we will get derailed. I’ll make a separate post on Field soon]
Vectors
An element having both magnitude and direction are called vectors.
Scalars
Scalars only possess magnitude.
Space
A space ,in general sense, is a collection of elements of similar behavior or activity bound in that region. If we consider our planetary space which is basically an accumulation of celestial bodies or a vector space in this regard is a collection of vectors which obey some condition to maintain this configuration.
Now that our premise is set, let’s dig down the definitions and understand it’s significance.
What is Vector Space?
A vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.
•The operation + (vector addition) must satisfy the following conditions:
•Closure: If u and v are any vectors in V, then the sum u + v belongs to V.
•(1) Commutative law: For all vectors u and v in V, u + v = v + u
•(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w
•(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V, 0 + v = v and v + 0 = v.
•(4) Additive inverses: For each vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V, called an additive inverse of v, and denoted by — v. The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:
•Closure: If v in any vector in V, and c is any real number, then the product
• c · v belongs to V.
•(5) Distributive law: For all real numbers c and all vectors u, v in V,
•c · (u + v) = c · u + c · v
•(6) Distributive law: For all real numbers c, d and all vectors v in V,
• (c+d) · v = c · v + d · v
•(7) Associative law: For all real numbers c, d and all vectors v in V, c · (d · v) = (cd) · v
•(8) Unitary law: For all vectors v in V, 1 · v = v
So any space of vectors following this particular configuration and obeying the conditions can be claimed as a Vector Space.It can be a linear map or a vector space R2 is represented by the usual xy plane etc.
Below we have discussed a few important properties regarding dependency of a vector space.
A FEW PROPOSITIONS ON DEPENDENCE AND INDEPENDENCE OF A VECTOR SPACE:
•A superset of a linearly dependent set of vectors in a vector space V over a field F is linearly dependent. (cases of the linearly dependent set being finite and infinite)
•A subset of a linearly independent set of vectors in a vector space V over a field F is linearly independent.(just as the previous ones)
•A set of vectors containing a null vector ϴ in a vector space V over a field F is linearly dependent.
•The set consisting of a single non-zero vector a in a vector space V over a field F is linearly independent.
•Deletion Theorem
In the next post I will talk about dependence and independence of vector space along with some trivial concepts. Don’t forget to show me your support by liking this post and comment your valuable feedback!
For reference I consult
Linear Algebra: A geometric Approach by S. Kumaresan
Linear Algebra Done Right by Sheldon Axler
BBYE!
HOPE TO MEET YOU SOON WITH THE NEXT POST!
@ PhiWhyyy!?!❤
