Extract 8.6
Context: Same college, tutor and students as Extract 8.1 (in the order 2, 1, 3, 4). The mathematical problem in all sessions is the following:
Given two vector spaces V and W and a linear mapping T: V®W, find matrix A for T.
The tutor suggests the following method:
Consider a basis {ei} for V and a basis {fj} for W.
Express T(ei) as combinations of fj.
The coefficients of these combinations form the columns of matrix A.
The Episode:
Considering a Basis for P3(Â). As an application of the tutor's method (see Context), the students and the tutor find the matrix for
T: P3(Â)®P3(Â) where Tp(x)=p(x+1).
All agree that, because V=W, it can be ei=fj and that dimP3(Â)=4. In Tutorials 2 and 3 the students give the 'usual' basis, as the tutor calls it, {ei, i=1,...,4}= {1, x, x2, x3} immediately. In tutorials 1 and 4 the students seem to have difficulties. In the following, I present the discussion that the tutor's request to the students to give the 'usual basis' in Tutorials 1 and 4:
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P1: Is it...1+x....oh no... The tutor asks for 'simpler' polynomials. To her question 'how p(x) looks like' Frances replies 'a x cubed plus b x square plus c x... plus d'. So p(x) is a linear combination of? P2: x3, x2, x and the constant. T: Well... ? P3: Just one. |
Beth gives '1 0 0 0, 0 1 0 0... '. The tutor reminds her that they are talking about polynomials, not matrices. To her silence the tutor asks 'how p(x) looks like' and Beth replies 'a x cubed plus b x square plus c x... plus d' where a,b,c and d are real numbers. So p(x) is a linear combination of? B1: Er,... to the s from 1 to 3... The tutor insists that they are 'looking for some simple looking polynomials. Nice simple things' and repeats her question about what 'simple polynomials' is p(x) made out of. B2: Mmm... it's just made by linear ones... T: What do you mean linear ones? B3: It's a product of three... The tutor realises that Beth is talking about factorisation and she stresses that multiplication of polynomials is not linear. Asking Beth to 'take a step back' she repeats that they are trying to write p(x) as a linear combination of four other polynomials d1,..., d4 which form a basis. She asks Beth what is a linear combination of d1,...,d4 ? B4: ad1+bd2+cd3+dd4... The tutor says she does not approve of this notation for the coefficients but moves on and asks Beth to compare ad1+bd2+cd3+dd4 with ax3+bx2+cx+d Beth then dictates the 'usual basis' {1, x, x2, x3} for P3(Â). |
Express T(ei) as combinations of ei. The tutor asks the students to calculate T(1). The discussion in the four tutorials is as follows:
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P4: Two. The tutor says that she knows what Patricia has done. They laugh and the tutor brings their attention to the definition of T. Patricia says it is x+1 and the tutor wonders what happens if there is no x to map. To their silence she notes that 'in fact there is nothing to do'. Patricia exclaims 'ah!' and says that then T(1) is 'just one'. |
The students' initial silence is followed by Abidul's suggestion ('1+x') for which she can however give no reason. The tutor repeats that we have to replace x with x+1 and asks what 'problem they have with it'. A1: x doesn't appear. ... One? |
C1: Two. The tutor disagrees and asks what 'problem they have with it'. 'Take your polynomial and wherever x appears, replace it with x+1' she suggests. Cleo: Just one.
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B5: Polynomial 2. Asked what she means she replies: B6: Because we are adding one. The tutor repeats the definition of T and notes that to add 1 you first have to have an x. B7: Is it one? |
Subsequently the tutor prompts the students with closed questions so that they give the coefficients of T(1) and thus construct the first column of A ((1, 0, 0, 0)). The students immediately follow with the other three columns. In the following, the discussion is from Tutorial 3.
While calculating Cleo asks what would happen if instead of x they had 1+x. The tutor says that Cleo has made 'a good point' and asks the students whether 1, 1+x, (1+x)2, (1+x)3would form a basis too. Camille nods and asks if matrix A would then be the diagonal matrix. The tutor nods and says that generally different bases produce different matrices. Camille asks in what cases the matrix is the diagonal matrix. The tutor replies that there are ways of choosing the basis so that the matrix is the diagonal one. Camille asks whether this is because we choose T(di) as the basis. The tutor recommends cautious application of the method dictated by the proof of the Rank and Nullity Theorem: Camille recalls that you consider a basis for kerT and extend it to a basis for the domain of T. Then the images of the elements of this basis will form a basis for ImT. Cleo then asks 'What's this Im?', the tutor gives the definition of Image and closes by explaining that all the matrices of a mapping produced from different bases are of the same rank