Extract 8.3
Context: Student Connie has difficulty with 'bases and spanning sets'. Her confusion, she says, 'became bigger' in this morning's session — the group tutorial that this tutor teaches to the eight students of this college on the morning of their individual tutorial day. I note that in that tutorial the tutor suggested they replace the expression åaivi from the definition of
the span of a set S={v1, v2,..., vk}ÍV as <S>= {åaivi / aiÎF} where F is the field over which the vector space V is defined
with the expression 'linear combinations of'. Connie had used this notation and the tutor recommends caution because å implies the 'infinite sum'. In the Linear Algebra course they will only need the finite sum of vectors to express the elements of an infinite set. The span of a finite set can be an infinite set. For example, on the plane the span of S={(0,1)}, which is a finite set, is the x-axis. More generally if S<V (S is a subspace of a vector space V) then <S>=S.
The Episode:
The tutor has just said that if S<V then <S>=S. A rather puzzled Connie asks hesitantly if 'this is because of the axioms'. Specifically because of closure under addition and scalar multiplication, the tutor adds. He stresses that in the Linear Algebra course of this year they are restricted in the cases where SÍV and S is finite. He then asks for her intuitive idea on <{(1,0), (1,1)}>.
C1:...lamda (1,0)...
T1: What about geometrically?
C2: x-axis...
T2: What else?
C3: And the reals... does this become... constant if you take this as zero...?
T3: Yes, but what about geometrically if you do that? What vectors in the plane do you take?
C4: You take any...
The tutor says he agrees with C4: the span of these two vectors is the whole plane and this is the smallest number of vectors that span the plane. These two vectors form a basis for the plane and there is a theorem according to which all bases have the same number of vectors. This number we call the dimension of the space.
Connie then asks him to complete the proof that he started in the morning tutorial that
for a set SÍV where V is a vector space <<S>>=<S>.
The tutor presents the proof (in brief: <S>£ V and <T>=T when T£V). She then asks him to explain why in LA5.24
rowspace (PA)£rowspace(A)
for P an rxm matrix and A an mxn matrix.
The tutor starts by defining rowspace(A)=<{ri}> as the span of the set of columns ri of matrix A. Connie interrupts to ask:
C5: Is the spanning set the set of all... all the... same property about image... it's all about the values that thing can take?
The tutor advises Connie not to think in terms of THE spanning set since for a subspace there are various spanning sets. He suggests using the expression 'a set being spanned by S'.
C6: So that means it spans something... so it's the values that it takes...
The tutor defines <S> as the set of all the finite linear combinations of the elements of S and reminds her of the plane example. She then asks:
C7: Why are they [spanning sets] useful?
Spanning sets are not particularly interesting, he replies; bases are, because they provide with a finite expression of the elements in a vector space. Also a dimension is a characteristic of the vector space and there are important theorems involving the dimension of a vector space. Connie interrupts the tutor in order to return to LA5.24.
C8: Basis is like... from the question here... when we have a spanning set... you use row operations to find a basis in there... and one of them disappears... so is the basis the smallest set of... the smallest...
The tutor repeats that she should avoid expressions such as THE basis of a vector space. He then points at various bases of the plane.
Connie then asks how LA5.24 can be used to prove LA5.25, as requested in the sheet. The tutor explains that by reducing the matrix of the vectors that span U to echelon form, which is what is suggested in LA5.24, we can find the minimum number of vectors that span U. These vectors will form a basis of U and here the reduction to echelon form leads us to a minimum number of 2. 'So is this the basis?' wonders Connie (C9) and the tutor repeats that this is one basis for U.
C10: So is this a 2D basis?... I can't understand intuitively what it is.
The tutor reminds her that as they have always said in the Geometry lessons the plane is two-dimensional and the space is three-dimensional. That need to express vectors in space with three coordinates, or two on the plane, is an intuitive idea of a basis.
C11: And also x and y and r-thetas are also bases?
T4: r-thetas are another story. That's polar coordinates. Think about different bases of this type.
C12: So it's just a way of describing the axes.
T5: Right. If you like... it's a way of describing coordinates.
C13: So this is kind of a funny coordinate system.
The tutor agrees and repeats that there is a variety of bases for a vector space and that the technique he outlined above provided one of them.
Return to Appendices for Chapter 8.