Any mathematical endeavor starts with a set of assumptions/axioms/rules to define the boundaries of mathematical object being studied. In Group theory, a Group is a set of actions that abide by below rules:
- There is a predefined list of actions that never changes.(generators)
- Every action is reversible.
- Every action is deterministic.
- Any sequence of consecutive actions is also an action
Sub group is a group that exists inside a group. We generally write it as B<A.
Normal subgroup is a subgroup who’s all left cosets are equal to it’s right cosets.
Normalizers of subgroup H which is not normal in Group G is NhG which represents set of all generators whose left coset of H is equal to right coset of H.
Isomorphism is special case of Homomorphism
How can one say or verify that a subjective conscious experience of an objective real world entity is necessarily the same or different for different people? Though everyone identifies a particular wavelength of light as red(word, not conscious experience itself), It is entirely possible that different individuals might have different conscious experience for same entity and can identify it with same notation or word(red) as an interface for communication.
If we assume that different individuals necessarily have same conscious experience for the same entity, then what is it in our mind that constitutes to same experience for similar objective entities in many different individuals of evolutionary sprouts?
Perception is not possible with mind and no matter (or) with matter and no mind. Hence Perception is an emergent property from the interactions of mind and matter. Lot has been already said and discussed in physics about matter.
P=>Q is a statement. It says If P is true, then Q is also true.
If you find a case were P is true and Q is false, then the statement P=>Q lied to us which means it is false.
For all the remaining cases of T T , F T , F F , P=>Q isn’t necessarily falsified.
Just think of truth table of implication as a list of all possibilities in the universe of A and B and when do the implication statement fails to satisfy.
Students try to compare Matrix multiplication to normal multiplication first time they hear or see matrix multiplication and they get lost without developing any intuition for what does it actually mean.
Now everybody knows that normal multiplication b*a just means adding a for b times, don’t get me started with the case of non-integers here, because we will deal with that in other article.
Normal multiplication is kind of dealing with vector(a) in one dimension i.e. straight line. You can go anywhere (or span) the whole line just by multiplying it with varied values of b.
Suppose vector(a) is two dimensional of sort(xi+yj). Since the vector is two dimensional by it’s very nature, it has the potential to span whole two dimensional space if we multiply it by right kind of b. If b is scalar, then you are stuck in one dimension i.e. in the direction of xi+yj. If b is a two dimensional vector, then multiplying b with col(xi yj) gives you another column vector. ooh.. slow down buddy.. what has happened just now..Think of two columns of two dimensional vector as two vectors, if x and y or 1,2 then you take 1 times first column vector, then 2 times second column vector and add both of them, result is another column vector. This is called linear combination. This is not really different from the concept of vector where you take x times in i direction and y times in j direction to end up at a point in 2-d except for the fact that you are taking x times in 1st column vector direction and y times in 2nd column vector direction. Matrix multiplication is Linear Transformation or transformations based on the size of matrices being multiplied.
Think of column vectors of the multiplication matrix , often left side one in A*B , as the places where original i,j,k.. unit vectors of your n dimensional space end up after matrix multiplication. Now think of your multiplication result of A*B as the places where column vectors of B end up after the Linear transformation or Matrix multiplication.
Matrix Multiplication is non-abelian i.e. AB!=BA
Hope this is helpful..Correct me if I am wrong.