6874
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2011-07-04T14:55:09
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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. These transformations can have very clear, intuitive properties and we can often think of them from either a geometrical perspective or an algebraic perspective.<br></br>
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Below are ten questions about the properties of such transformations in three dimensions for you to think about.<br></br>
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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?<br></br>
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<ol>
<li>What does a matrix do to the zero vector ${\bf 0}$?</li>
<li>What does a matrix do to a line/plane through the origin?</li>
<li>What does a matrix do to a line/plane not through the origin?</li>
<li>Which lines can you transform onto the x-axis using matrix multiplication?</li>
<li>Which planes can you transform onto the xy-plane using matrix multiplication?</li>
<li>Can you think of a matrix which transforms a plane to a line?</li>
<li>Can you think of a matrix which transforms a line to a plane?</li>
<li>How many matrices transform the cube $(\pm 1, \pm 1, \pm 1)$ to another cube?</li>
<li>Can you find a matrix which transforms a square to a triangle in 2D?</li>
<li>Can you think of a matrix which shifts all points from ${\bf x}$ to ${\bf x+ (1,0,0)}$?</li>
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<p><span class="editorial">We have not yet received any solutions to this problem! Can you be the first?</span></p>
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<mdoxml version="1.0"><h3>Why do this problem?</h3>
<div>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.</div>
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<h3>Possible approach</h3>
<div>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.</div>
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<div>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.</div>
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<div>Some students may need reminding about the form of the vector
equations of a line and a plane.</div>
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<div>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.</div>
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<h3>Key questions</h3>
<div>Can you give an algebraic example to justify your answer to
the question?</div>
<div>How does a geometrical interpretation of the situation
help?</div>
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<h3>Possible extension</h3>
<div>Questions 8-10 are a little more challenging than the first
few questions in the problem.</div>
<div><a href="http://nrich.maths.org/6876&part=">Matrix
Meaning</a> extends students' understanding about the effects of
matrices which reflect or rotate in two or three dimensions.</div>
<h3>Possible support</h3>
<div><a href="http://nrich.maths.org/6875&part=">Square
Pair</a> gives a good starting point for considering how matrices
transform vectors in two dimensions. </div>
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The equations for a line and plane in vector form may be
useful.<br></br>
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Line: ${\bf r}={\bf a} + \lambda{\bf b}$<br></br>
Plane: ${\bf r}={\bf a} + \lambda{\bf b}+ \mu{\bf c}$<br></br>
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It may also be useful to recall that matrix multiplication is
distributive:<br></br>
${\bf M}({\bf a} + {\bf b}) = {\bf Ma} + {\bf Mb}$<br></br>
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<span style="font-style: italic;">This problem uses concepts met in
the later Further Pure Maths A
level modules.</span><br></br>
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<p> </p>
<div style="margin-top: 0px; margin-right: auto; margin-bottom: 0px; margin-left: auto; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; font-family: Georgia, 'Times New Roman', Times, serif; font-size: 12pt; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(255, 255, 255); width: 680px; color: rgb(34, 34, 34);">
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">1. $\pmatrix{a_1&a_2&a_3\cr a_4&a_5&a_6\cr a_7&a_8&a_9}\pmatrix{0\cr 0\cr 0} = \pmatrix{0+0+0\cr 0+0+0\cr 0+0+0} = \pmatrix{0\cr 0\cr 0}$ </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">A matrix transformation takes the origin to the origin.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">2.<strong style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </strong>$M(\lambda \mathbf{a} + \mu \mathbf{b}) = M(\lambda
\mathbf{a}) + M(\mu \mathbf{b}) = \lambda (M\mathbf{a}) + \mu (M\mathbf{b}) $</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">(Where $M$ is any matrix, $\mathbf{a}, \mathbf{b}$ are fixed vectors, $\lambda , \mu$ are scalars. $\lambda \mathbf{a} + \mu \mathbf{b}$ defines a plane. If $\mathbf{b} = \mathbf{o}$, this expression defines a line.)</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">Planes through the origin are transformed into other planes or lines through the origin or a point at the origin by matrix transformations. Lines are transformed into lines through the origin or a point at the origin.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">3.<strong style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </strong>$M(\lambda \mathbf{a} + \mu \mathbf{b} +
\mathbf{c}) = M(\lambda \mathbf{a}) + M(\mu \mathbf{b}) + M\mathbf{c} = \lambda (M\mathbf{a}) + \mu (M\mathbf{b}) + M\mathbf{c}$</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">(Where $M$ is any matrix, $\mathbf{a}, \mathbf{b} \mathbf{c}$ are fixed vectors, $\lambda , \mu $ are scalars)</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">Planes not through the origin are transformed into planes or lines or points, through or not through the origin.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">4. Any line - think about matrices of the form $\pmatrix{a_1 & a_2 & a_3 \cr 0 & 0 & 0 \cr 0 & 0 & 0}$</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">5. Any plane - think about matrices of the form $\pmatrix{a_1 & a_2 & a_3 \cr a_4 & a_5 & a_6 \cr 0 & 0 & 0}$</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">6. The example given for part 4 transforms everything into a line!</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">7. The result from part 2 shows this to be impossible.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">8. Any combinations of enlargements, rotations and reflections. These are the only shape-preserving transformations.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">9. Matrix transformations take parallel lines to parallel lines (or points) - think about the result of part 3, and the equations of parallel lines.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">10. No - the origin does not move under a matrix transformation</p>
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