What do you mean by this? Do you mean writing a vector ##v=\sum_{\iota \in I}v_\iota b_ \iota## as ##v=(..., v_\iota , ...)## in coordinate form according to a basis ##\{b_\iota\}_{\iota \in I}##?Silviu said:
A vector is something which a matrix applies to: ##v \mapsto Av##. Of course you may take ##v## as a row or column vector of ##A## and fill up the rest with zeroes, but why? ##v## and ##A## are different objects, one is something that make up vector spaces and the other one is a mapping between vector spaces.
which means ##Mv = c\, v##.
So your vector ##v## (with the dimensions above) is a ##(3\times 1)-##matrix and ##M## a ##(3\times 3)-##matrix.
This depends on how you "represent" ##v## by ##A##. If ##A = (v,0,0)## then of course. But why should you do this?
Again. There is no meaning in "representing" a vector as a square matrix, unless in very special cases (which I can't imagine). The only natural way is to see a vector as a ##(n \times 1)-##matrix.
Hello! Thank you for your answer! The reason why I am confused is what I read in a book about SU(2) representation. I attached a part of what I read. They aim to "reduce the Hilbert space of the world to block diagonal form". I am confused as the Hilbert space contains vectors, but block diagonal is related to matrices. This is why I assumed they try to represent vectors by square matrices... Do they mean something else there?fresh_42 said:
Hello! Thank you for this, but I am still a bit confused. They say "the states of the representation can be written as (j, m)". What do they mean by the state of a representation? I thought that the state (j, m) is an element of the vector space, while the representation is a matrix representation of the group of operations acting on this vector space. But, by what they are saying, I understand that they represent the vector space itself, and I am a bit confused why would you do that in general? (I understood the reason for representations of the group of operations, but I can't understand yet the reason of representing the vector space where you apply the operations).fresh_42 said:
I have found a description of this connection (between math and physics language) on the example of Pauli matrices here. It's in the wrong language, but you could either ignore the texts and only regard the formulas or take it as bad written English since the important words are almost the same. (I've linked to the relevant section, so it's not necessary to consider the entire page. It's been easier than to retype all formulas.) I haven't looked into deep but I hope it can help you. Also the English version of this page (which isn't simply a translation) could be interesting to read.Silviu said:
A vector is a mathematical object that has both magnitude (size) and direction. It is represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude.
A vector can be represented using various methods, such as a column or row matrix, a numerical list, or using coordinates on a graph. The most common representation is using a coordinate system, with the x and y axes representing the direction of the vector.
A scalar is a single numerical value, while a vector is a combination of magnitude and direction. Scalars can be represented by a single value, while vectors require multiple values to fully represent them.
Vector representation is important in many fields of science, such as physics and engineering, as it allows us to accurately describe and calculate quantities with both magnitude and direction. Vectors are also essential in understanding concepts such as velocity, force, and acceleration.
Vector addition and subtraction can be represented geometrically by placing the vectors tip-to-tail and drawing the resulting vector from the origin to the tip of the final vector. It can also be represented algebraically by adding or subtracting the corresponding components of the vectors.
