That is already a first and most important "why". It may be a little bit too terse, but mathematics didn't evolve by knowing the applications before doing something.
Without doing some playing around from sqrt(-1) and discovering how it connects one thing to another, nobody is able to come up with real applications. You need to at least build a placeholder of the concept in your mind before you can examine what's possible.
So a person with a similar mindset as those who were the first people to use complex numbers, would just try to find a way to express a square root of a negative number and see how it goes. It starts with a limitation of an important tool and tries to close a perceived conceptual gap. The mathematicians themselves that write this book probably all think that way. I wouldn't call myself a mathematician but I didn't need anything more than that sentence to believe someone was motivated enough just from that reason alone.
So really there's a whole audience out there - arguably that a professional mathematician most wants to address - that could appreciate this sentence just as-is. So it's not true that it's natural to think the audience requires further explanation. Whether you should care is another matter. But as this is a book about linear algebra not complex numbers, some others would have accused the author of digression if he granted your wishes.
So I don't think what you're demanding is fair. Maybe it's a reasonable request after the fact, but it's a little too harsh to think it's something the author must have addressed in his text to his intended audience. This kind of inquiry is what in-person teaching is useful for.
This is completely false, there is a 'why' and it's that people needed to permit taking square roots of negative numbers to find (real!) roots of cubic polynomials. I don't think this book needs to digress into that, sure.
Without doing some playing around from sqrt(-1) and discovering how it connects one thing to another, nobody is able to come up with real applications. You need to at least build a placeholder of the concept in your mind before you can examine what's possible.
So a person with a similar mindset as those who were the first people to use complex numbers, would just try to find a way to express a square root of a negative number and see how it goes. It starts with a limitation of an important tool and tries to close a perceived conceptual gap. The mathematicians themselves that write this book probably all think that way. I wouldn't call myself a mathematician but I didn't need anything more than that sentence to believe someone was motivated enough just from that reason alone.
So really there's a whole audience out there - arguably that a professional mathematician most wants to address - that could appreciate this sentence just as-is. So it's not true that it's natural to think the audience requires further explanation. Whether you should care is another matter. But as this is a book about linear algebra not complex numbers, some others would have accused the author of digression if he granted your wishes.
So I don't think what you're demanding is fair. Maybe it's a reasonable request after the fact, but it's a little too harsh to think it's something the author must have addressed in his text to his intended audience. This kind of inquiry is what in-person teaching is useful for.