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, ..., vn are linearly independent (l.i.), if and only if:
- c1v1 + c2v2 + ... + cnvn = 0
has only the trivial solution (ie ci=0).
Question What does this say about pivots for the augmented matrix for the
Connection to Efficiency/Redundancy
If a set of vectors is
not l.i. then at least one ci 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
is the set of all linear combinations
of these vectors. In other words, every b in span can be written as
c1v1 + c2
v2 + ... + cnvn
for some constants ci.
A collection of vectors is a basis if it both spans and is
linearly independent, ie represents the space efficiently.