Extract 8.8
Context: Two pairs of female students are taught in two tutorials about finding the matrix of a mapping between two vector spaces with respect to some particular bases:
Tutorial 1: Patricia and Cleo
Tutorial 2: Beth and Cary.
In Tutorial 1 a previous discussion made evident some of the students' difficulties with row operations. In the following, the tutor and the students discuss B3. Some of the students did some work on it but none of them completed the question. B3 (solution provided in fig.8) is about
Proving that E' is a basis of V
Finding the matrix of T with respect to bases E' and F
Finding a basis F' such that a given matrix is the matrix of T with respect to bases E' and F'.
The Episode:
Proving that E' is a basis of V. The students suggest proving that E' is a basis of V by proving that E' spans V and is linearly independent. The tutor notes the redundancy of their thinking: they only need to prove one of these conditions. The students agree but, when asked why, none of them can answer the question. Only Patricia vaguely replies 'you can deduce the one from the other' and, when asked to specify how, she wonders whether it is because we already know that E' is linearly independent. The tutor disagrees and points out that, since dimV=3 and E' has three elements, according to a theorem they have done, all they need to show is either that E' spans V or is linearly independent.
Finding the matrix of T with respect to bases E' and F. In Tutorial 2 the students have successfully found the matrix so no discussion of this part of the problem takes place.
In Tutorial 1 the students remain silent when the tutor asks them about the meaning of a matrix of a mapping T with respect to bases E and F. Then Cary says: 'That Te equals f. All the elements in it... ' but gives no further explanation. The tutor then asks specifically what Te1 is going to be. The students are silent and at some point Patricia points at another, irrelevant matrix on Problem Sheet B and whispers something like 'that's the matrix of the transformation'. The tutor repeats that her question generally concerned the matrix with respect to bases E and F and, in particular, their idea of what the rule is that 'connects the map with the matrix'. Patricia hesitantly points at the columns of the matrix and whispers: 'The coefficients of the sums... f1 plus f2... '. The tutor explains that 'the first column gives you the coefficients of Te1' and, at her request, the students dictate the calculations correctly.
Finding a basis F' such that the given matrix is the matrix of T with respect to bases E' and F'. In Tutorial 1 this part of the problem is not discussed. The tutor has asked them to complete the question on their own.
In Tutorial 2, the tutor and the students discuss their findings on basis F':
B1: They only have two elements.
T1: Oh, is that the proper conclusion to get to? What do you know about... ?
C1: I've got three. The third one is zero.
T2: Ah, can you put zero in a basis?
C2: [after a pause]... You can't.
T3: Right. Because... think about the zero vector...
C3: Oh, do... these can be written...
T4: Yes, it's not linearly independent... your three vectors will be linearly dependent... and what's the dimension going to be?
B2: Two.
T5: Ah, but what this... if you have a basis f1, f2, f3...
B3: [after a pause]... Then you have to have three...
T6: Right,... OK, so Cary is right: you've got to find a third one. Em,... what... do you know the first two? [Silence] When you are constructing f' how does your reasoning go about what you should put in it?
B4: Te'1 will be f'1... and Te'2 has got to be f'2...
T7: Right. And we know that in any case that Te'3 is... ?
B5: Zero.
T8: Zero. So we need... so what condition do we need on f'3?
B6: [after a pause]... To equal zero.
T9: No. We said that it mustn't equal zero.
B7: It's got to be linearly independent with f'1 and f'2.
T10: Does it matter what it is?
B8: No.
The tutor stresses that 'you can make f'3 whatever you please as long as you make it linearly independent with these two' and Beth chooses f'3 = f1. The discussion closes with a check on linear independence and with the tutor's generalisation from 3x3 matrices to mxn.
Return to Appendices for Chapter 8.