> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn.
If you read the book in the original post you may find it's absolutely for you.
Axler assumes you know only the real numbers, then starts by introducing the commutative and associative properties and the additive and multiplicative identity of the complex numbers[1]. Then he introduces fields and shows that, hey look we have already proved that the real and complex numbers are fields because we've established exactly the properties required. Then he goes on to multidimensional fields and proves the same properties (commutativity and associativity and identities) in F^n where F is any arbitrary field, so could be either the real or the complex numbers.
Then he moves onto vectors and then onto linear maps. It's literally chapter 3 before you see [ ] notation or anything that looks like a matrix, and he introduces the concept of matrices formally in terms of the concepts he has built up piece by piece before.
Axler really does a great job (imo) of this kind of bridge building, and it is absolutely rigorous each step of the way. As an example, he (famously) doesn't introduce determinants until the last chapter because he feels they are counterintuitive for most people and you need most of the foundation of linear algebra to understand them properly. So he builds up all of linear algebra fully rigorously without determinants first and then introduces them at the end.
[1] eg he proves that there is only one zero and one "one" such that A = 1*A and A = 0 + A.
If you read the book in the original post you may find it's absolutely for you.
Axler assumes you know only the real numbers, then starts by introducing the commutative and associative properties and the additive and multiplicative identity of the complex numbers[1]. Then he introduces fields and shows that, hey look we have already proved that the real and complex numbers are fields because we've established exactly the properties required. Then he goes on to multidimensional fields and proves the same properties (commutativity and associativity and identities) in F^n where F is any arbitrary field, so could be either the real or the complex numbers.
Then he moves onto vectors and then onto linear maps. It's literally chapter 3 before you see [ ] notation or anything that looks like a matrix, and he introduces the concept of matrices formally in terms of the concepts he has built up piece by piece before.
Axler really does a great job (imo) of this kind of bridge building, and it is absolutely rigorous each step of the way. As an example, he (famously) doesn't introduce determinants until the last chapter because he feels they are counterintuitive for most people and you need most of the foundation of linear algebra to understand them properly. So he builds up all of linear algebra fully rigorously without determinants first and then introduces them at the end.
[1] eg he proves that there is only one zero and one "one" such that A = 1*A and A = 0 + A.