Why do this problem?
This problem gives a good opportunity to think about the
action of matrices on vectors in geometrical as well as algebraic
terms, allowing students to develop ideas about what multiplication
of a vector by a matrix actually means.
One approach that works well is to divide the class into
groups and give each group some of the questions to work on. A
possible grouping is questions 1-3, 4-7, and 8-10, with 8-10 being
the most challenging.
Ask each group to first read through each question and decide
whether they have any intuitive feel for what the right answer
might be. Then they should use algebra and/or geometrical arguments
to justify their answers. In cases where the answer depends on
various factors, students should clearly explain what these factors
are. Explain that at the end of the session they will have to
justify their answers to the rest of the class, so they should
prepare a presentation to explain their findings.
Some students may need reminding about the form of the vector
equations of a line and a plane.
At the end, allow plenty of time for students to present their
answers to the questions they were given, and encourage the rest of
the class to be critical, asking questions and challenging anything
that doesn't make sense to them.
Can you give an algebraic example to justify your answer to
How does a geometrical interpretation of the situation
Questions 8-10 are a little more challenging than the first
few questions in the problem.
extends students' understanding about the effects of
matrices which reflect or rotate in two or three dimensions.
gives a good starting point for considering how matrices
transform vectors in two dimensions.