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**Problem**

This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions.

**Relevant Example**

Take [itex]SO(3)[/itex] for example; it's the group of [itex]3\times 3[/itex] orthogonal matrices of determinant [itex]1[/itex] under matrix multiplication. We can represent elements of the group as [itex]3 \times 3[/itex] rotation matrices, so it makes sense to interpret the group itself as a rotation group in [itex]\mathbb{R}^3 [/itex]. The rotation preserves the volume and radii of a collection of points. What I don't understand is the meaning of a representation of this group in terms of matrices of dimension other than [itex]3[/itex]. The definition of the group itself seems to assume [itex]3 \times 3[/itex] matrices, so how do matrices of higher dimension make sense, and what is their interpretation? Are they still rotations in [itex]\mathbb{R}^3[/itex] ? What are the preserved quantities?

**What I Know**

I get that [itex] SO(3) [/itex] has three generators because there are three continuous parameters needed to define the group, and that as long as they satisfy the commutation relations you can have any representation you want (so, matrices of a higher dimension are mathematically allowed) but I can't figure out what the other representations

*mean*. I also understand that, for example, different representations of [itex]SU(2)[/itex] correspond to different spins, but that just seems more abstract.