Compare the span of the 3 vectors Vector([1,0]), Vector([1,1]) and Vector([0,1])
to the span of the 2 vectors Vector([1,0]) and Vector([0,1])
a) The spans are the same and I have a good reason why
b) The spans are the same but I am unsure of why
c) The spans are different but I am unsure of why
d) The spans are different and I have a good reason why



How to express the redundancy? 1.7:

The vectors v₁, v₂, ..., v_n are linearly independent (l.i.), if and only if:

c₁v₁ + c₂v₂ + ... + c_nv_n = 0 has only the trivial solution (ie c_i=0).

Question What does this say about pivots for the augmented matrix for the system?



Connection to Efficiency/Redundancy If a set of vectors is not l.i. then at least one c_i is nonzero, say it is c₁, and then v₁ is a linear combination of the other vectors. If a set is l.i. then no vector is redundant, as in throwing any away would span a smaller space, and so the full set is an efficient set in this sense.

The span of a collection of vectors v₁, v₂, ..., v _n is the set of all linear combinations of these vectors. In other words, every b in span can be written as

b = c₁v₁ + c₂ v₂ + ... + c_nv_n for some constants c_i.

A collection of vectors is a basis if it both spans and is linearly independent, ie represents the space efficiently.