#   Math 2270
# Maple Project 2
# October 14, 20002. 
# 
# A Maple text version of this project may be found at the web site
# http://www.math.utah.edu/~kapovich/teaching11.html/
# Go to that page and click on the "2-nd maple assignment".  
# I recommend saving the file from the website onto you home directory,
# and then
# opening it from Maple, as a ``Maple text'' document.  You can then do
# the assignment by inserting the proper commands as well as italicized
# textual comments in answer to the various questions.  In order to
# execute multiline commands from the version you get from the web, you
# should use the ``Edit'' option in your maple window to ``join''
# execution groups which you have highlighted with your mouse.  
# 
# I remind that in each problem you should use Maple 
# to solve the problem, not pen-and-paper computations! 
# 
# PART 1. 
# 
# In the first part of the assignment we will study the geometric
# meaning of linear (hence matrix) maps
# f(x)=Ax, from R^n to R^m.  We will focus on maps from R^2 to R^2 to
# illustrate more general properties.  
# Of course, computer graphics are most concerned with those maps and
# with R^3
# rotation maps and projection maps from R^3 to R^2.  
#     
#  The very definition of a linear map, namely that f(u+v)=f(u)+f(v) 
#  and f(su)=sf(u), for all vectors u,v and scalars s, has the following
#  geometric consequences:
# 
#  Lines are mapped to lines, and parallel lines are mapped to
# parallel lines:  This is because a line can be described as a set 
# L={u + t v ,  where  t  is in R}, where u is a point on the line and v
# is 
# a direction vector.  Therefore the image set  
# f(L):={f(u+tv)}={f(u)+tf(v)}  
# is also a line, going through f(u) with direction f(v).  (If the
# directions
# f(v) turns out to be 0, then the line degenerates into a point.) 
# 
# Therefore, line segments are mapped to line segments,  polygons are
# mapped to polygons, and regions bounded by polygons are mapped to
# regions bounded by polygons; if we know where the vertices go, we know
# everything.  Let's use these facts to draw the images of some
# polygonal regions under a matrix map.
# 
#     First load the linear algebra and plotting libraries
# 
> with(plots):with(linalg):
> 
> verts0:=[[0,0],[1,0],[1,1],[0,1]]; 
> 
>       #corners of unit square
> 
> unitsquare:=polygonplot(verts0, color=`yellow`):
> 
>       #this command make a polygon and colors
> 
>       #the region inside it yellow.  Make sure
> 
>       #to end this command with a colon!
> 
> display(unitsquare);  #Now semicolon! this command
> 
>       #shows the square
> 
# When you executed the sequence of commands above you should have
# gotten a picture of a yellow unit square.  Now we will use a linear
# map with matrix A defined below to map this square to a parallelegram
# 
> A:=matrix([[3,2],[-1,1]]);  #the matrix of our
> 
>        #random linear transformation
> 
> f:=x->evalm(A&*x);  #our linear map
> 
# Problem 1. 
# 
# 1a)  Assignment:  For our map f(x)=Ax defined above, what are the
# images of the
# points e1=[1,0] and e2=[0,1]?  How do you find these images from the
# rows or columns of A?
# 
# We use the ``map'' command below to see where the vertices of the unit
# square are sent by f.  The syntax is to put the mapping function in as
# the first argument, and the list of input points as the second
# argument. The result of the command  will be the list of output
# points.  Check (not to hand in) that this is what happens below to
# the four points in the list verts0. 
# 
# Note that here we are conflating points and vectors in R^2, namely we 
# identify each point  P  with its coordinates, i.e. with the vector
#   -->
#   OP
# 
# 
> verts1:=map(f,verts0);
> 
> image1:=polygonplot(verts1, color=`red`):
> 
> display({unitsquare,image1});
> 
# 1b)   Assignment: Explain where f(e1) and f(e2) are represented in the
# picture you 
# just made.  (Mark them by hand on the picture.)  
# 
# By the way, if you click on the plot you can choose to
# have the projection ``constrained '', in which case Maple will use the
# same scales for the vertical and horizontal axis.  Otherwise it scales
# the x and y-axes to make the picture fit nicely onto your screen.
# 
# 1c)  Assignment: If we compute f(f(x)):=f^2(x), then what will the
# matrix be for
# the resulting linear transformation? After answering that question,
# make the vertices for f(f(unitsquare)) as follows, and draw the 
# corresponding image polygons.
# 
> verts2:=map(f,verts1);
> image2:=polygonplot(verts2,`color`=blue):
> display({unitsquare,image1,image2});
# 
# 1d)   Assignment: Explain what the columns of the matrix for f^2 have
# to do with
# the picture you just made above.
# 
# 1e)   Assignment: What is the inverse function to f ?  (Hint : what is
# its matrix? ) 
# Verify that the inverse mapping takes image1 back to the unit square
# by using the method
# from 1c). 
# 
# 
# Problem 2.
# 
#  Translations of objects are mapped to translations of the image 
# objects and scalings of objects are mapped to scalings of the image
# objects:  
# 
#     First let's review  what do we mean by an ``object'' and
# by translations and scalings of an object.  An object  is some set   S
# 
# of points  s.   By the  image of   S  we mean the collection of image 
# points   f(s) .  We write  f(S)   for this image (like we did for the
# line 
# L and its image  f(L)  in problem 1).   Similarly if   b   is a
# translation
# vector, then the object  S+b  means all points of the form  s+b  where
#  s
# is in  S.  If  c  is a scalar, then the scaled (or dilated) set  cS 
# means
# all points of the form  cs , where  s  is in  S.
# 
#     Now, if we apply the linear map f to the translated object S+b we
# get all points of the form f(s+b)=f(s)+ f(b), where  s  is in  S
# (since   f  is 
# linear), i.e. exactly the set  f(S)+f(b) , which is the translation of
# 
# the image  f(S)  by the vector  f(b).  Similarly, if we apply  f  to
# the 
# scaled set cS  we get  f(cS) to be the set of all points  f(cs)=cf(s) 
# (since f is linear), i.e. the scaling  cf(S) of the image set  f(S).
# 
#      Here's how to translate and scale the unit square from #1:  
# Let's translate it by the vector b=[2,3]:
# 
> b:=[2,3];
> 
> trans:=x->evalm(x+b);
> 
> verts3:=map(trans,verts0);
> 
> image3:=polygonplot(verts3,color=`yellow`):
> 
>       #colon!
> 
> display(unitsquare,image3);
> 
# Here's how to scale it by a factor of 0.2:
# 
> c:=0.2;
> 
> shrink:=x->evalm(c*x);
> 
> verts4:=map(shrink,verts0);
> 
> image4:=polygonplot(verts4,color=`red`):
> 
> display({image4,unitsquare});
> 
#  
# 
# 2a)  Assignment: Describe where you expect the translated unit square,
# image3
# above, to be  mapped  by our linear map  f  from problem 1.  Then use
# the ``map'' command and the ``trans'' command, as well as polygonplot
# and display, to make a picture of the images of the unit square and
# its translation when they are mapped by  f .  Verify that the images
# differ by the expected translation.
# 
# 
# 2b)   Assignment: Describe what you expect the image of the scaled
# down square
# above to be when you apply  f , and then draw (using MAPLE!) a picture
# illustrating this.
# 
# 
# Problem 3.   Special linear transformations in R^2 (rotations,
# reflections, projections, shears):
# 
# 3a) Rotations:  The matrix for rotating by an angle theta is:
# 
> Rot:=theta->matrix([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]])
> ;
> 
# so to rotate the vector (2,3) by Pi/3, we would command
# 
> theta:=Pi/3;
> 
> evalm(Rot(theta)&*[2,3]);
> 
> 
> 
# Assignment for 3a): Draw a picture of the unit square rotated by Pi/3
# radians.
# 
# 3b)  Reflections (see section 2.2 of the  textbook).
# 
# Assignment:   Define matrices which give reflections across the 
# x-axis, across the line y=-2x, and across the line y=x.  
# Illustrate what each of these linear maps does to the unit square (as
# we did in 1a).  
# Finally, verify (using MAPLE computation) that  reflecting across the
# line y=x is the 
# composition of first rotating by  Pi/4  in the clockwise direction,
# then reflecting across
# the x-axis, then rotating (back ) Pi/4 in the counterclockwise 
# direction. 
# 
# Hint: if L: x --> y and  M: y--> z are linear transformations 
# and N: x--> y=L(x) --> z= M(y)  is their composition, and the matrix  
# of  L  is A, the matrix for M   is   B, then the matrix for N is A&*B.
#   
# 
# 
# 3c)  You can scale by different factors in the x and y-directions. 
#  Assignment: define a  matrix expands the x-direction by 3 and the 
# y-direction by 2? (This is an analogue of the "Procustes"
# transformation discussed in the class).  
# Make a picture of what this transformation does to the unit square
# similarly to 1a). 
# 
# 3d)  Projections.   Assignment: Write down the matrix which projects
# points onto 
# the line y=2x.  Illustrate what this projection map does to the unit
# square.  Is there an inverse map? 
# 
# 3e) Shears:  Recall that a shear of strength   k   in the x-direction
# is defined by the matrix
# 
> Shear:=k->matrix([[1,k],[0,1]]);
> 
#  Assignment: For k=1 explore what the shear map does to the unit
# square. What 
# happens if you apply the shear map twice?  Three times?  Make pictures
# using MAPLE. 
# 
# 
# PART 2. 
# 
# This part of the project is about the fundamental subspaces associated
# with
# matrices and about coordinates with respect to different bases. 
#   
# Here's the matrix we will work with.  We will keep it moderately
# sized, but of course we could make it much bigger without causing
# Maple any problems. 
# 
> A:=matrix([[1,1,1,2,6],[2,3,-2,1,-3],[3,5,-5,1,-8],[4,3,8,2,3]]);

                         [1    1     1    2     6]
                         [                       ]
                         [2    3    -2    1    -3]
                    A := [                       ]
                         [3    5    -5    1    -8]
                         [                       ]
                         [4    3     8    2     3]

> b:=vector([0,0,0,0]);

                          b := [0, 0, 0, 0]

# We will think of this as the matrix of a linear transformation 
> L: R^5-> R^4# ,
>   L(x)=Ax.#  
# 
# Problem 4.  Assignment: Compute rank and nullity  of A using reduction
# 
# of A to row echelon form. 
# 
# 
# Problem 5.  Assignment: Find a basis for the kernel of A. 
# To solve this problem you can either use the "rref" command or
# "linsolve": 
# 
> linsolve(A, b);
# In this command  A  is the matrix,  b  is the vector 
# of the right-hand side (in the present problem you will use "zero"
# vector).
# If you do not want to use the symbols like _t_1 as your parameters 
# you can use the command:
# 
> linsolve(A, b, 'r', s);
# Now the names of the parameters will be s_1, s_2,... Type 
# >r;
# to see what the rank of the matrix is. 
# 
# Problem 6.  Assignment: Use Maple to find a basis v1, v2, v3 for the
# column space of A 
# (i.e. of the image of  L).  Recall that one way to find this basis is
# to do
# row operations, putting your matrix into reduced column echelon
# form.  
# 
# Problem 7.  Maple will compute bases for these spaces with  single
# commands. 
#  Assignment: Compare the answers you've gotten above with Maple's
# answers:  Since
# you know bases are not unique, you can pretty well guess that Maple is
# using methods close to the ones we used, since its answers should be
# quite similar to some of yours:
# 
> rowspace(A); # Gives a basis for the space spanned by row vectors.
> nullspace(A);#nullspace basis
> colspace(A); #nice column space basis, i.e. a basis for the image of
> L. 
> v:=convert(colspace(A),list); 
# #The above command is making a list of vectors in the basis. This
# keeps track of order in the set v 
# #of vectors which form basis of the image of L.  
>     v1:=v[1]; v2:= v[2]; v3:= v[3] 
# These vectors form a basis of the image of L. Compare these vectors
# with vectors from the problem 6.  
> r1:=row(rref(A),1); #r_1,r_2,r_3 basis for rowspace(A)
> r2:=row(rref(A),2);
> r3:=row(rref(A),3);
> G:=convert(nullspace(A),list); #making a list
>     #keeps track of order in a set
> n1:=G[1];n2:=G[2]; #nullspace basis. # 
# 
# Problem 8.  Assignment: Use Maple to verify that the vectors r1, r2,
# r3, n1, n2 form a basis of R^5. 
# Hint: use reduced row echelon form or the determinant. 
# Explain your solution: namely, how the value of the determinant or
# shape of the reduced echelon form is 
# related to the fact that we have a basis. Check that each of the
# vectors r1, r2, r3 is orthogonal to each of the vectors n1,n2. 
# I.e. the row-space  is orthogonal to the kernel  of the matrix. 
# 
# Problem 9.   Assignment: Let us call E={e1,e2,e3,e4,e5}, i.e. the set
# of standard basis
# vectors in R^5.  Let us call our new basis  S={r1,r2,r3,n1,n2}. 
# Find the transition matrices 
> P_{E <-S}#  and 
> P_{S<-E}# , which 
# convert S coordinates to E-coordinates and vice-versa. Recall that the
# matrix 
> P_{E <-S}#  
# is very easy to find and the matrix 
> P_{S<-E}#  is easily obtained from the matrix 
> P_{E <-S}.  
# 
# Problem 10. Using transition matrix find the coordinates of the
# following vectors with respect to
# the S-basis:
# 10a)  a=(0,1,-4,0,-3),
# 10b)  b=(1,0,0,0,0).
#