Describing vectors in n dimensions

[Nano-Passion]

Recall that A can be broken up into its components x,y, and z. Can You simply add more components to describe any number of dimensions. Where n would be nth dimension?

A=A_x+ A_y+A_z+⋯A_n

 

[BruceW]

What's the square symbol for? And why do you have theoretical physics degree in your status thingy? Do you mean you will someday hope to get such a degree? I'm hoping of doing a theoretical phd someday. I didn't know there was such a thing as a purely theoretical undergraduate degree, if that's what youre talking about?

About the question: If we have a vector A, then it can be written (in some basis) as its components (A₁,A₂,A₃,A₄) For a four-dimensional vector, so yeah, there are as many components as there are dimensions.

 

[Nano-Passion]

What's the square symbol for? And why do you have theoretical physics degree in your status thingy? Do you mean you will someday hope to get such a degree? I'm hoping of doing a theoretical phd someday. I didn't know there was such a thing as a purely theoretical undergraduate degree, if that's what youre talking about?

About the question: If we have a vector A, then it can be written (in some basis) as its components (A₁,A₂,A₃,A₄) For a four-dimensional vector, so yeah, there are as many components as there are dimensions.

For PhD I wrote what I'm striving for. Perhaps that isn't what I should do. I'll edit it.

I've only begun my third semester. :)

Edit: What square symbol?

 

[BruceW]

A=A_x+ A_y+A_z+⋯A_n
The symbol ⋯ just before A_n it looks like two horizontal lines..

 

[Nano-Passion]

A=A_x+ A_y+A_z+⋯A_n
The symbol ⋯ just before A_n it looks like two horizontal lines..

Oh sorry, that is supposed to be an ellipse (...) that indicates the sequence.

 

[WannabeNewton]

You don't write it that way. The components are relative to a basis as Bruce mentioned. A vector can be represented as a linear combination of components relative to said basis vectors so it would be written compactly as V = V^{\alpha }\vec{e}_{\alpha } where the repeated indices imply summation over all possible values the index can take. So, for example, a vector V \in \mathbb{R}^{n} can be written, when equipped with the Cartesian chart, as V = V^{x}\vec{e_{x}} + ... + V^{n}\vec{e_{n}} where \vec{e_{x}} = (1, 0, ..., 0) and similarly for the other basis vectors.

 

[Nano-Passion]

You don't write it that way. The components are relative to a basis as Bruce mentioned. A vector can be represented as a linear combination of components relative to said basis vectors so it would be written compactly as V = V^{\alpha }\vec{e}_{\alpha } where the repeated indices imply summation over all possible values the index can take. So, for example, a vector V \in \mathbb{R}^{n} can be written, when equipped with the Cartesian chart, as V = V^{x}\vec{e_{x}} + ... + V^{n}\vec{e_{n}} where \vec{e_{x}} = (1, 0, ..., 0) and similarly for the other basis vectors.

I have to admit, I am not familiar with that notation. But I will mention that my book stated A=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}. Which is why I simply put:

A=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}+...A_n\widehat{t}

 

[WannabeNewton]

I have to admit, I am not familiar with that notation. But I will mention that my book stated A=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}. Which is why I simply put:

A=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}+...A_n\widehat{t}

No that is perfectly correct. It just wasn't in your original post is all.