To me, one of the interesting things about how mathematical results have been discovered historically is that they have often been accidental (or intended for something else).
eg. eigenvalues were originally looked at for use with quadratic forms, but people later found that its interesting properties (orthogonality, symmetry, etc) were useful for lots of other things.
There is often also a large gap in time between when a mathematician playing around with a problem discovers an interesting result and when it is actually applied to something useful.
eg. Euler's law regarding complex exponentials was developed around 1740, but it wasn't until around 1807 when Fourier used it in harmonic analysis and 1897 when Steinmetz started applying it to electrical engineering.
But you wouldn't know that from reading a textbook, which is (understandably) arranged in such a way that omits historical context and why people bothered to study it in the first place. Most linear algebra books are classic examples of how to introduce abstract topics without any context.
eg. eigenvalues were originally looked at for use with quadratic forms, but people later found that its interesting properties (orthogonality, symmetry, etc) were useful for lots of other things.
There is often also a large gap in time between when a mathematician playing around with a problem discovers an interesting result and when it is actually applied to something useful.
eg. Euler's law regarding complex exponentials was developed around 1740, but it wasn't until around 1807 when Fourier used it in harmonic analysis and 1897 when Steinmetz started applying it to electrical engineering.
But you wouldn't know that from reading a textbook, which is (understandably) arranged in such a way that omits historical context and why people bothered to study it in the first place. Most linear algebra books are classic examples of how to introduce abstract topics without any context.