Mark44 said:
A basis is a set of vectors that is
1)linearly independent
2)spans the space or subspace it is found in.
I don't think this is an improvement over the OP's definition. The set ##\{(1,0,0)\}## "is found in" (I can only assume that this means "is a subset of") infinitely many subspaces of ##\mathbb R^3##, but it only spans one of them. If you meant that the set spans the space it spans, then the second statement isn't saying anything.
The OP's definition is fine in my opinion. If I had to change something about it, I would add some clarity by mentioning the space for which the set is supposed to be a basis:
Let ##V## be a vector space. A set ##B\subseteq V## is said to be a
basis for ##V## if
(a) ##B## is linearly indendent.
(b) ##B## spans ##V##.
MarcL said:
I'm just having a small trouble understanding the difference
[...]
Anybody got a good way to differentiate both?
Let ##V## be a vector space. Let ##S## be a non-empty subset of ##V##. Let ##W## be a subspace of ##V##. The following statements are equivalent:
(a) ##W## is the intersection of all subspaces of ##V## that has ##S## as a subset.
(b) If ##U## is a subspace of ##V## such that ##S\subseteq U##, then ##W\subseteq U##. (In other words, ##W## is the
smallest subspace that contains ##S##).
For each non-empty subset ##S##, we define the
span of ##S## as the unique subspace ##W## that satisfies the equivalent conditions above. Different books use different notations for this subspace. Some common ones are ##\operatorname{span} S## and ##\bigvee S##. The set ##S## is said to
span ##W##, to
generate W, and to be a
spanning set for ##W##. (These statements all mean the same thing, that the equivalent conditions above are satisfied).
Let ##V## be a vector space. Let ##S## be a subset of ##V##. The following statements are equivalent.
(a) ##S## is a maximal linearly independent set in ##V##.
(b) ##S## is a minimal spanning set for ##V##.
If you're not familiar with the minimal/maximal terminology, then these statements need to be explained. This is what they mean:
(a) For all ##T\subseteq V##, if ##T## is linearly independent, then ##T\subseteq S##.
(b) For all ##T\subseteq V##, if ##T## spans ##V##, then ##S\subseteq T##.
One book I read used this theorem to define the term "basis". Such a definition would look like this:
Let ##V## be a vector space. A subset ##S\subseteq V## is said to be a
basis for ##V## if it satisfies the equivalent conditions of the theorem. (In other words...if it's a minimal spanning set for ##V##, or equivalently, a maximal linearly independent set in ##V##).
MarcL said:
To determine whether or not a set spans a vector space, I was taught to find its determinant and if det|A|=/= 0 then it spans the space.
I was also taught that if det|a|=/=0 then it isn't coplanar and therefore it is linearly independent ( can also just solve to see if trivial sol'n...)
But then if both det|a=/= it means that it spans and is linearly independent. Therefore, in my head, it comes with the idea that "spans is related to independence"
This approach can't be applied to sets like ##\{(1,0,0),(0,1,0)\}\subseteq R^2##, because the matrix ##\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\end{pmatrix}## isn't even square. To get a square matrix when you consider a subset of ##\mathbb R^n##, you have to start with a set with ##n## elements. If you find that the determinant is non-zero, this tells you that you have found a linearly independent set. Since it has ##n## elements, and ##\mathbb R^n## doesn't contain
any linearly independent subset with ##n+1## elements, it's a
maximal linearly independent set, and therefore a basis.