> This is true, but it's also something that is arguably impossible on the one hand and "unwise" on the other. If I understand "unwise" to refer to a linear ma

> This is true, but it's also something that is arguably impossible on the one hand and "unwise" on the other. If I understand "unwise" to refer to a linear map into a lower-dimensional space: of course that's something you'll often want to do! Suppose, for example, that most of the interesting structure of your data is close to lying on a n-dimensional subspace of R^m, with n<m. Constructing a clever linear map from R^m to R^n can be very wise and useful! > And in the other direction, you can't actually map a lower-dimensional space into a higher-dimensional space with a matrix. The image of X can never have more dimensions than X does -- the choice to represent it with Y > X dimensions is just that, a representational choice. Any matrix involves a choice (of basis), so that complaint is moot. For sure you can have a linear map whose codomain is higher-dimensional. You are correct that the image can't be higher-dimensional, but that doesn't prevent the existence of such a map.