Is a scalar different than a 1x1 matrix or vector?

Question

Is a scalar different than a 1x1 matrix or vector?

Answer 1

1x1 Matrix

Answer 2

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Hello friend MY CITY, time is definitely scalar. Because there is no need to have a direction concept for counting the time. TIme itself is a peculiar concept. Time is actually nothing but the change. Change is relative to the observer. So time is a relative concept. Same way most of us think that current has to be a vector. But current is a scalar quantity though we mark the direction of current in a circuit. The main reason is that current is defined as the rate of flow of charge. So I = q / t As charge is having only magnitude it is a scalar. Now right from I = q / t, we get q = I t Now already current is a scalar. Charge is scalar. So t, the time, has to be scalar. One more important point is that we assign + and - for charge. It does not give any assurance for the charge to be a vector. These sign are known as algebraical sign. As we add +5 C and -6 C we get only -1 C. So algebraic addition of charges. More over current density is a vector quantity. Why? Current density J = I / A As current I cannot be a vector, how can current density become a vector? I = J A Product of J and A gives scalar. This recalls the scalar product of two vectors. So A ie area has to be a vector. Is area a vector??????!!!!!!! Yes area can be considered as vector. How is it possible? Take a circle. Its area is ie magnitude is pi r^2 But how can we bring the direction into the scene? Circle has got two faces. As you look at one face it may be taken to be positive. So the other side is to be considered as negative. Recall vector product of length and breadth of a rectangle. L x B Length and Breadth can be considered as vectors. As L to B is anticlockwise then LxB is taken to be positive. If L to B goes in clock wise then LxB will be negative. So area becomes a vector. So scalar product of current density and area vector gives the current which is scalar. Thank you very much for having given me a nice platform to express my thoughts towards these interesting information

Answer 3

A scalar quantity can have at most one single dimension. It MAY have a direction ( forward and backwards) but no more than that. So use these rules. Can time have more than one dimension or just a single one? Can time have a value in either the forward or backward direction but nothing other than this? If so then it may be considered to be a scalar. A vector has multiple independent dimensions where there are many conditions in which 1+1 does NOT equal 2 ( nor zero either). i.e 1 North + 1 east does not equal 2 in any direction. Only if we artificially constrain the movement along a single track ( railway track?) then the movement can be considered as a scalar.

Answer 4

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RE:

Is a scalar different than a 1x1 matrix or vector?

A website I just stumbled across on matrix algebra started their discussion with vectors. Now I already know vector algebra, but I went to this site for its matrix discussions. They said that two vectors, an [nx1] and a [1xn], have two different types of products: an outer and an inner. In my...

Answer 5

My friend cheeser tells me that he knows the answer to this question, and that again you are missing out by being arrogant and cranky when the mathematical facts he presents you with turn out to be in conflict with your mixed-up intuition.



He says that, briefly, that you are doing matrix multiplication wrong. The inner product is:



      [ 1 x n ] [ n x 1 ]



The outer product is:

      [ n x 1 ] [ 1 x n ]



One result in a 1x1 matrix which we identify as a scalar. The other is an nxn matrix. Remember that dimensions of marices multiply as [ m x n ] [n x k ] = [ m x k]



Cheeser also tells me that you are right, at least in thinking that scalars are "the same" as 1x1 matrices. He suggests you look up a "vector space isomorphism" and consider that the isomorphism sending scalar k to matrix [k] is such a thing.



And please, we all know you think he is a "douche" for daring to tell you that in order to "define" Fibonacci numbers over all reals, you should use the exponential formula involving the golden ratio (even though it still fits most facts about the Fibonacci sequence, including f[n+2] = f[n+1] + f[n]).



So don't bother ranting about him any more. We've all heard it.

Answer 6

Matrix multiplication is not generally commutative. That is AB ≠ BA. For example a 3x1 column vector A, times a 1x3 row vector B, is a 9 element matrix, while BA is a 1x1 matrix or scalar.

Answer 7

A scalar is the same as a 1 x 1 matrix or vector.



You're right, the rules of matrix multiplication will give you an n x n matrix in the case of multiplying a [n x 1] by a [1 x n] matrix, in that order, and will give you a 1 x 1 matrix in the other order.



I don't understand your point about "removal of the scalar from the matrix". As you say, all of this is just applying the rules for matrix multiplication consistently.