Numerical linear algebra has a long and vast history. You really do need specialized information that your undergrad linear algebra classes would not have imparted.
For instance, the condition number is important:
This is really good book.
It contains various topics from basic linear algebra (like solving systems with LR factorization) to solving (partial) differential equations with linear algebra. It also tells you where each method applicable (like, "you can use that if this, but that one method works almost everywhere you could need, except that it is much more expensive").
Russian translation of that book quickly became a rarity.
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