Lecture 28: Subsets, Subspaces, Linear Combinations, Span (Nicholson Section 5.1)

Alternate Video Access via MyMedia | Video Duration: 46:27  

Description: Introduce language “subspace” and “span”.  

1:45 --- Introduced 6 subsets of R^2 using set notation.

4:48 --- Presented each subset as a collection of vectors in R^2. Note: I was not as careful as I could have been --- when presenting one of the quadrants (Q2, for example) I simply shaded in a region of the plane as if it were a collection of points. Really, I should have drawn one position vector for each point in that region, filling up the quadrant with infinitely many vectors of varying lengths and angles (but with the angles always in a certain range, determined by the subset Q2).  

21:30 --- Defined what it means for a subset of R^n to be a subspace of R^n.

22:00 --- Went through the previously introduced 6 subsets and identified which were subspaces.

24:50 --- Introduced a subset S of R^4; is it a subspace? Proved that it’s a subspace of R^4.

33:44 --- Given a set of vectors in R^3 --- if one lets S be the set of all linear combinations of these vectors, what would this set look like in R^3?  

35:20 --- another subset of R^3. Is it a subspace?

37:27 --- a subset of R^2. Is it a subspace?

42:45 --- another subset of R^2. Is it a subspace?

43:35 --- Given {v1, v2 ,.. vk} a set of k vectors in R^n, introduced Span(v1, v2 ,.. vk) and stated that it’s a subspace of R^n.