Based on the additive white quantization noise model, linear transform coders are derived for Gaussian sources corrupted by noise. There are two alternative design objectives: minimizing the trace of the error correlation matrix and thus minimizing the mean-squared error, or minimizing the determinant of the error correlation matrix and thus maximizing information rate. It is shown that a solution to both problems is to first transform the noisy observations into canonical coordinates, quantize and apply a Wiener filter in this coordinate system, and then transform the result back to the original coordinates. Canonical coordinates are uncorrelated, and quantization and Wiener filtering are applied to each component independently. The type of canonical coordinate system depends on the design objective: Quantization in half-canonical coordinates minimizes the mean-squared error and quantization in full-canonical coordinates maximizes information rate. Finally, it is also demonstrated in this paper that majorization is the fundamental principle underlying proofs of optimal transform coding.