Advanced Linear Algebra, Lecture 1.2: Spanning, independence, and bases

Advanced Linear Algebra, Lecture 1.2: Spanning, independence, and bases

A subset S of a vector space X is a spanning set if every vector in X can be written as a linear combination of elements in S. It is linearly independent if there is only way to write the zero vector. Finally, it is a basis for X if it spans and is linearly independent. Loosely speaking, this means that it is "big enough to generate", but "not too big as to have redundancies". We introduce these concepts and prove some basic results, including that any two bases have the same size. This leads to the definition of the dimension of a vector space.

[dongxuli]

Great channel. I found linear algebra is the actual field I have to revisit from time to time. Thanks a lot.

[mpoullet]

Slide 3: "where the intersection is taken over all subspaces of X that contain *S* ."
Slide 4: *y_2* =b_1*y_1+b_2*x_2+...+b_n*x_n

[魏寅生]

I like the reminder of group theory part

[udaydeshpande]

One way to say proof for the "if" clause is: Span of S is a subspace itself which contains all linear combinations of S and nothing else and hence is one of the Y-alphas. Any proper subset of Span S will necessarily exclude some linear combinations of S and hence won't remain a subspace. So span of S is the least Y-alpha. All other Y-alphas must at least contain span of S since they all contain S. So the intersection of all Y-alphas must be span of S. Now of course, one needs to write this in maths.

[mohammedal-haddad]

One of my favourite channels. Thank you very much.