MRI signal acquisition with multiple coils in a phased array is nowadays commonplace. The use of multiple receiver coils increases the signal-to-noise ratio (SNR) and enables accelerated parallel imaging methods. Some of these methods, like GRAPPA or SPIRiT, yield individual coil images in the k-space domain which need to be combined to form a final image. Coil combination is often the last step of the image reconstruction, where the root sum of squares (rSOS) is frequently used. This straightforward method works well for coil images with high SNR, but can yield problems in images with artifacts or low SNR in all individual coils. We aim to analyze the final coil combination step in the framework of linear compression, including principal component analysis (PCA). With two data sets, a simulated and an in-vivo, we use random projections as a representation of the whole space of orthogonal projections. This allows us to study the impact of linear compression in the image space with diverse measures of reconstruction accuracy. In particular, the $L_2$ error, variance, SNR, and visual results serve as performance measures to describe the final image quality. We study their relationships and observe that the $L_2$ error and variance strongly correlate, but as expected minimal $L_2$ error does not necessarily correspond to the best visual results. In terms of visual evaluation and SNR, the compression with PCA outperforms all other methods, including rSOS on the uncompressed image space data.