Wavelets, as their name implies, are little waves. More specifically, they are oscillatory functions that increase and decrease.
Wavelets first began to be studied in the 1930s. They were an offshoot of Fourier analysis. Fourier analysis is a representation of a function as a series. These series can then be used to analyze signals, much like wavelets. In the 1930s, wavelets were found only in the realm of pure math. However, by the 1980s, many real-world applications had been found for wavelets.
You can find applications for wavelets in many diverse fields. As of 2010, they are used in image compression, quantum physics, signal processing and seismic geology. The wavelets can be used to describe how an electric signal behaves.
Linear algebra is usually studied as an undergraduate. If you are a math major, it falls into the range of pure mathematics. You will not only be studying lines, but all functions whose exponential value is one. Vectors and matrices are important components of linear algebra.
You can study the wavelet transformations by using the inner product in linear algebra. The resulting vectors can then be stored in a matrix and used for various applications.
Wavelets can be a rather complicated concept, difficult for the undergraduate to understand. By introducing them through linear algebra, it gives you the chance to assimilate the basic idea of this complex concept with an idea you have already studied.