You can skip to "The thing that is bothering me:-" if you want to read only question.
Background:
Probably this question has been asked in the past but I am having some difficulty please bear with me. So, I will tell you what I've learned so far and then we will come to part I cannot understand. I am studying a Mathematics MOOC on Algebra and other concepts related to data analysis.
Question:
The thing that I know:-
A vector is used to represent properties of an entity or any measurable value or just some point in space or some direction, it is part of broader concept called tensors, it is a tool of convenience for mathematical operations. A matrix is also a tool for vector operations, the operation I'm interested in is division or reduction, matrix's inverse is used to reduce a matrix to Identity matrix because a matrix cannot be divided, it is used to find unknown variables in system of "linear" equations or it can be used to find a vector in new coordinate system or vector space using:-
New Basis vector[my coordinate systems] * My Vector == Vector[new coordinate System]
This formulae is independent of orthogonality of basis vectors. If it were orthonormal we could have used vector projection to find our new vector. Now to determine that if a matrix is inversible we need to find it's determinant which should not be zero(0). This is also used to determine if vectors of a given matrix are dependent. Dependency is used to find the span of a vector space or if a vector can be written as combination of other vectors.
The thing that is bothering me:-
When we multiply a matrix with it's inverse the number of 1's in it's identity matrix represent the span of matrix, right? What happens when we reduce a transformation matrix like in equation above to find an unknown vector? And suppose it is a 4x4 transform matrix but after reduction it has one less number of 1's in it's identity matrix, How does it affects our vector? What new transform looks like? Is there a loss of data? How would you define determinant as a mathematical concept? Any suggestion about some free online tool where all these concepts can be visualized?