Transformations for 10
Problem
The operation of mutiplying a vector by a constant matrix can by thought of as transforming a point in space onto another point in space.
Below are ten questions about the properties of such transformations in three dimensions for you to think about. There are some hints and suggestions in the Getting Started section.
As you think about the questions, can you draw relevant diagrams and construct relevant algebraic examples? In each case, is there a definitive answer, or does it depend on various factors? You may intuitively feel the answers to some of these questions; in these cases can you prove your intuition correct?
- What does a matrix do to the zero vector ${\bf 0}$?
- What does a matrix do to a line/plane through the origin?
- What does a matrix do to a line/plane not through the origin?
- Which lines can you transform onto the $x$-axis using matrix multiplication?
- Which planes can you transform onto the $xy$-plane using matrix multiplication?
- Can you think of a matrix which transforms a plane to a line?
- Can you think of a matrix which transforms a line to a plane?
- How many matrices transform the cube $(\pm 1, \pm 1, \pm 1)$ to another cube?
- Can you find a matrix which transforms a square to a triangle in 2D?
- Can you think of a matrix which shifts all points from ${\bf x}$ to ${\bf x+ (1,0,0)}$?
There are more matrix problems in this feature.
Getting Started
You can start by thinking about what happens to the origin under a matrix transformation by considering $\pmatrix{a_1&a_2&a_3\cr a_4&a_5&a_6\cr a_7&a_8&a_9}\pmatrix{0\cr 0\cr 0}$.
Where must the origin map to under a matrix transformation?
Can more than one point map to the origin under a matrix transformation? For example, you could consider the multiplication $\pmatrix{1 & 0 & -1\cr 0 & 1 & 1\cr 1 & 1 & 0}\pmatrix{x\cr y\cr z}$ and find some values of $x, y$ and $z$ which are mapped to the origin.
The equations for a line and plane in vector form may be useful.
Line: ${\bf r}={\bf a} + \lambda{\bf b}$
Plane: ${\bf r}={\bf a} + \lambda{\bf b}+ \mu{\bf c}$
Note that if ${\bf a} = {\bf 0}$ then the line or plane passes through the origin.
It may also be useful to recall that matrix multiplication is distributive:
${\bf M}({\bf a} + {\bf b}) = {\bf Ma} + {\bf Mb}$, and also that we have ${\bf M}(\lambda {\bf a}) = \lambda {\bf M}{\bf a}$.
You might like to consider what happens to parallel lines by considering ${\bf M}(\lambda {\bf b} + {\bf a_1})$ and ${\bf M}(\mu {\bf b} + {\bf a_2})$. What does this imply must happen to a square?
Teachers' Resources
Why do this problem?
Possible approach
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.
Key questions
Can you give an algebraic example to justify your answer to the question?
Possible extension
Questions 8-10 are a little more challenging than the first few questions in the problem.
Possible support
There are more matrix problems in this feature.