Linear Span
link of original video: Linear combinations, span, and basis vectors $\vert$ Chapter 2, Essence of linear algebra
2-D
- "linear" in linear combination: (y=ax_1+b_x2) if we fixed one of those scalars, and let the other one change its value freely, the tip of the resulting vector draws a straight line.
- the set of all possible vectors that you can reach with a linear combination of a given pair of vectors is called the span of those two vectors.
- the span of most pairs of 2-D vectors is all vectors of 2_D space, but when they line up, their span is all vectors whose tip sits on a certain line.
- the span of two vectors is basically a way of asking, "what are all the possible vectors you can reach using only these two fundamental operations, vector addition and scalar multiplication?"
- the point at the tip of the vector, where we can think about its tail sits on the origin.
3-D
- the idea of span gets a lot more interesting if we start thinking about vectors in 3-D space… If we take two vectors in 3-D space, that tip will trace out some kind of flat sheet, cutting through the origin of 3_dimensional space. This flat sheet is the span of the two vectors, or more precisely, the set of all possible vectors whose tips sit on that flat sheet is the span of our two vectors.
- we want some terminology to describe the fact that at least one of these vectors is redundant - not adding anything to our span. Whenever this happen, where you have multiple vectors and you could remove one without reducing the span. the relevant terminology is to say that they are "linear dependent". (it's already in the span of the others).
- on the other hand, if each vector really does add another dimension to the span, they are said to be "linearly independent".
The basis of a vector space is a set of linearly independent vectors that span the full space.