Question
Answered step-by-stepAsked by mike12gd on coursehero.com
. 5. [HAND IN 12 marks] In the diagram below, the 2x2 shaded...
Image transcription text
5. [HAND IN 12 marks] In the diagram below, the 2x2 shaded square centred at the origin is bounded by
the lines x = +1 and y = +1. Let T denote the linear transformation that maps the square on the left to the
parallelogram at the right.
T
(a) Use the structure theorem for linear transformations to find the matrix of T. Check your work with a
suitable point, e.g. x = [1, 1].
Image transcription text
[b] Now write Tas a product of three transformations, firsta dilation D, second a horizontal shear S, and
third a rotation R about the origin. Find matrices for each of the three transformations, and check that
their product gives you the matrix of T. Now there are many ways to do this, both graphical and algebraic. Here's how we want you to do it. It's
mostlya graphical argument The keyidea is to track the top of the box through the three steps. The point
is that if you know where the top of the box is, you know everything because the image of the box is
always a parallelogram centered at the origin. After the dilation the top of the box is still horizontal but it has a different length and a different height
After the shear, the top of the box is still horizontal with the same length and height, but it's shifted
horizontally. Finally, since the diagram tells you where it is after the rotation, you will be able to work out
the angle of rotation, and then you can continue [working backwards] from there. That is, first find R, and
then S, and then D. Show your work and on the template [next page] make an accurate drawing of the parallelogram after
each stage D and S.
Answer & Explanation
Solved by verified expertAnswered by ProfessorMetalDugong25 on coursehero.com
<p>secte<span class="eqneditor-formula" data-value="x=\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix}"><span class="katex"><span class="katex-mathml"></span><span class="katex-html"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mrel">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">s</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">s</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">sec</span><span class="mord">s</span></span></span></span><span class="vlist-s">sec</span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em;"></span><span class="arraycolsep" style="width:.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">sec</span><span class="mord">s</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">s</span></span></span></span><span class="vlist-s">sec</span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">s</span></span></span></span></span></span></span></p><p>secte<span class="eqneditor-formula" data-value="T= \begin{bmatrix} 4 & -4 \\ -4 & 4 \end{bmatrix}"><span class="katex"><span class="katex-mathml"></span><span class="katex-html"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:.13889em;">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mrel">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">s</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">s</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">sec</span><span class="mord">s</span></span></span></span><span class="vlist-s">sec</span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em;"></span><span class="arraycolsep" style="width:.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">sec</span><span class="mord">s</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">s</span></span></span></span><span class="vlist-s">sec</span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">s</span></span></span></span></span></span></span></p><p> </p><p>sectetur adipiscing elit. Nam lacin<br/>sectetur adipiscing elit. Nam<br/>sectetur adipiscing elit. Na</p>
Unlock access to this and over
10,000 step-by-step explanations
Have an account? Log In
Step-by-step explanation
<figure class="image"><img src="/qa/attachment/36937460/" alt="36937460"/></figure><figure class="image"><img src="/qa/attachment/36937474/" alt="36937474"/></figure><figure class="image"><img src="/qa/attachment/36937529/" alt="36937529"/></figure>sectetur adipiscing elit. Nam lacinia pulvinar tortor nec facilisis. Pellentes
3 attachments
JPG
JPG
JPG
Get unstuck with a CliffsNotes subscription
Unlock every step-by-step explanation, download literature note PDFs, plus more.Get Access