Using Linear Algebra for DSP
Yes indeed modern signal processing uses matrix methods of linear algebra (or Linear System Theory more correctly), in addition to the classical calculus and harmonic analysis.

Linear algebra (matrix) based DSP considers (1D) signals as vectors and transforms as matrices that map inputs to outputs. As CedronDawg also mentioned, in such a representation the DFT (discrete Fourier transform) is merely a matrix mapping:

$$ Y = \Phi X $$ where $X = [x_1,x_2,...,x_N]^T$ is an $N \times 1$ input vector, $Y = [y_1,y_2,...,y_N]^T$ is the output vector,  and $\Phi$ is the $N \times N$ DFT transform matrix.

For a brief treatment of matrix methods in DSP, have a look at the books:


Multiresolution Signal Decomposition_AKANSU
Statistical Digital Signal Processing_HAYES