Proving that all bases of a vector space have the same cardinality, and that any $n$ independent vectors of an $n$-dimensional space form a basis

Question

I am looking for literature that contains proofs of the following two theorems (we just wrote in class that it can be shown that they hold - for these two theorems):

It can be shown that all bases have the same cardinality.
It can also be shown that if $ n $ is the dimension of a vector space $ V $, then every linearly independent set with $ n $ elements is a basis for the vector space $ V $.

I managed to find literature for 1., but not for 2. Can someone suggest me a book, that maybe contains both?

If you know 1. and that every linearly independent set can be extended to a basis then you get 2. — Jochen, Aug 14 '24 at 07:49
If the set wasn't a base already you should be able to expand it so as to form a basis. But since all bases have the same cardinality we cannot add any further elements which means the set is a basis to begin with — Hari Shankar, Aug 14 '24 at 07:50
Related: If $n=\dim(V)$ and $n$ vectors are linearly independent, then they form a basis. — Anne Bauval, Aug 14 '24 at 08:06
Just for clarity, are you talking exclusively about finite dimensional vector spaces? Certainly point 2 suggests that (as $n$ usually denote a natural number, and also point 2 fails in infinite dimensional spaces), but point 1 could be considered in infinite dimension; the proof would be quite different though. — Marc van Leeuwen, Aug 14 '24 at 09:05

score 6 · Accepted Answer · edited Aug 14 '24 at 08:51

When it comes to the two theorems you mentioned, they are indeed fundamental results in Linear Algebra. Let me briefly walk you through where you can find detailed proofs and explanations in commonly used textbooks.

All Bases Have the Same Cardinality:
- This theorem tells us that any two bases of a vector space must have the same number of elements, and this number is what we call the dimension of the vector space. The proof generally involves the concept of the Steinitz Exchange Lemma, which is a crucial tool in linear algebra. You’ll find a detailed proof in textbooks like "Linear Algebra Done Right" by Sheldon Axler or "Introduction to Linear Algebra" by Gilbert Strang. These books present the material in a clear and structured way, making the proofs quite approachable.
A Linearly Independent Set with (n) Elements in a Vector Space (V) of Dimension (n) is a Basis:
- This theorem states that if you have (n) linearly independent vectors in a vector space (V) of dimension (n), these vectors form a basis for (V). The typical proof involves showing that these vectors span the entire space, meaning any vector in (V) can be expressed as a combination of them. This concept is central to understanding how vector spaces work, and again, you’ll find thorough explanations in the aforementioned books by Axler and Strang, as well as in "Linear Algebra and Its Applications" by David C. Lay.

These textbooks are standard in the field and will provide you with not only the proofs but also the necessary background to fully understand these important theorems. Happy studying!

Proving that all bases of a vector space have the same cardinality, and that any $n$ independent vectors of an $n$-dimensional space form a basis

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