Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained. CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université