I agree- the Discrete Fourier Transform can be easily explained in terms of basic linear algebra. And that explanation is usually included in an introduction-level DSP book. Both Oppenheim and Proakis have a section devoted to this view- understanding the DFT as matrix multiplication starts you on the road to the FFT algorithm.
But most ECE undergrads will first see the Fourier transform in a continuous setting. Then it isn't so much an issue of "basic linear algebra"- students have almost certainly never talked about the space of square integrable functions, the inner product on this space, a basis for continuous functions, what it means for an infinite collection of objects to be orthogonal, etc... That is, if they have even taken any linear algebra yet!
Students learn to associate "Fourier Transform" with "complicated math" and then are discouraged when they learn about the DFT. At least, that has been my experience when TAing signal processing classes.
But most ECE undergrads will first see the Fourier transform in a continuous setting. Then it isn't so much an issue of "basic linear algebra"- students have almost certainly never talked about the space of square integrable functions, the inner product on this space, a basis for continuous functions, what it means for an infinite collection of objects to be orthogonal, etc... That is, if they have even taken any linear algebra yet!
Students learn to associate "Fourier Transform" with "complicated math" and then are discouraged when they learn about the DFT. At least, that has been my experience when TAing signal processing classes.