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Advanced Linear Algebra Questions

Posted on 2010-02-08 17:33 By mjmoore (8)
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I'm trying to make the following question into a Maple TA question;
Find the orthogonal matrix R that diagonalizes the matrix A = .
Using question designer and then a maple graded response area, I can assess if the student has entered a valid matrix for R; however, the problem is that the student then has to enter something of the form <<1,3>|<-3,1>> as their answer. It's not that bad for a 2x2 matrix but once I get to 4x4's this is going to get really messy, and it's almost a given that someone will mess up the syntax.

Can anyone think of a way for students to enter their answer in a format that isn't so confusing? Or does anyone have any ideas on how to approach a question like this differently?

Thanks for your time and any advice,
Mike

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acer's picture

Matrix

Comment on 2010-02-08 18:13 from acer (Maple Rating 4 2069)

I may be in the minority here, but I prefer using the Matrix() constructor with its listlist argument for the data. For example, I find that I only very rarely get the syntax wrong for something like this,

Matrix([[1,3],[-3,1]]);

You might also consider guessing what the common syntax mistakes are with using the angle brackets, and perhaps even convert/accept typical mistakes. For example, you could convert, <<1,3>,<3,-1>> and give partial or full marks.

acer

longrob's picture

agree

Comment on 2010-02-09 02:04 from longrob (Maple Rating 2 114)

I agree with acer here. As a new student of Maple I find that using angle brackets is prone to error, but using square brackets is quite intuitive, even for a 4x4 matrix.

I see your point. Getting

Comment on 2010-02-09 11:55 from mjmoore (8)

I see your point. Getting the student to actually construct a matrix with the Matrix command would be easier for someone just learning to use Maple and not aware of the angle bracket notation. What I was trying to get at was that even with the Matrix(); command, at least 1 in 10 people are going to mess up with the syntax - an extra comma accidentally, or one missing bracket - and with the large number of students taking this course in the future, we run the risk of having a lot of remarking to do.

My goal is to get a format that the students can easily enter their answer into once they've done the 'on paper' work. Ideally they shouldn't have to enter a bunch of Maple syntax. For example I created a template with 16 response areas arranged in the shape of a matrix. This works for things like inverses when there is only one corresponding answer, but for something like diagonalization where the order of the vectors can be random and scalar multiples of those vectors are also valid answers, I can't do something like that. 

Mike

Doug Meade's picture

TA allows for more flexibility

Comment on 2010-02-09 13:17 from Doug Meade (Maple Rating 4 1135)

Your array of 16 response areas is probably the way to go, but I think you are unnecessarily limiting the types of problems for which this could be used.

If I were doing this, I would write code to collect the user's information from these cells as a matrix, and then use the relevant mathematics to check if this matrix is correct. Using your example of a diagonalizing matrix, this would involve multiplying the student's answer and the original matrix and checking that the result is diagonal.

Here's a simple implementatin of a test if a matrix is diagonal:

IsDiagonal := A -> ArrayTools:-IsZero( A - LinearAlgebra:-DiagonalMatrix( LinearAlgebra:-Diagonal( A ) ) );

Lastly, I would suggest that you might create a general array with, say, 36 response areas and have the instructions emphasize that students should fill in only those entries for the appropriate size of the response. (You could have two extra fields for the number of rows and columns, but I prefer the idea of having some areas left empty and modifying my answer checker to check that the student's answer has the correct shape and size.

Just a couple of thoughts that I hope are of some use,

Doug

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Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu

Collecting answers?

Comment on 2010-02-09 14:05 from mjmoore (8)

I had the same idea Doug, but I can't see how to implement it.

You said that you would write a code to collect the users information from all 16 seperate response areas, and then convert that to a matrix and use maple grading code to determine its correctness. Do you know of a way to do this? The IsDiagonal code you supplied would work for grading, but it's the data collection process that I'm struggling with.

I suppose my question now becomes is it possible for Maple TA to pass a users answer from one response area to another? It would also be interesting to know if it is possible to store a users response as a temporary variable that can be accessed in the grading code of another.

Thanks,
Mike

Doug Meade's picture

multiple response in a single TA question: possible or not?

Comment on 2010-02-09 16:51 from Doug Meade (Maple Rating 4 1135)

I have to admit that I am not, currently, a MapleTA user. I thought it was possible to have multiple response fields in a single question, but as I look over the documentation it appears my memory is faulty. Maybe you can do something with a multipart question, but I did not see anything explicit showing how this might be done.

Hopefully, someone else with more current working knowledge of MapleTA, and it's possibilities, will be able to point you in a good direction.

Doug

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Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu

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