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Matrix Manipulation Worksheet Start by loading the linalg (linear algebra) package to run matrix formula (don’t worry about the warning message.)
restart; with(linalg): Warning, the protected names norm and trace have been redefined and unprotected
Create our matrix with which we will manipulate our image. In this example, let’s double the height of the image and flip it across the y-axis. For this, (1,0) will move to (-1,0) and (0,1) will move to (0,2). Therefore, the matrix is as below.
A:=matrix([[-1,0],[0,2]]);
A :=
Let’s set up a list of lists, which can be plotted in Maple. Remember, this is an object of the type listlist, or a list of smaller lists (in this case, we have lists with two elements, the coordinates).
f1 := [[0,0], [0,5], [3,5], [3,4], [1,4], [1,3], [2,3], [2,2], [1,2], [1,0], [0,0]]; f1 :=[ [ 0 0, ], [ 0 5, ], [ 3 5, ], [ 3 4, ], [ 1 4, ], [ 1 3, ], [ 2 3, ], [ 2 2, ], [ 1 2, ], [ 1 0, ],[ 0 0, ]] Here’s how you plot a list of lists. The style=LINE draws lines between subsequent points. We duplicated the first point at the end so that the last line is drawn. plot(f1, style=LINE, color=blue, thickness=2, view=[-10..10, -10..10], scaling=constrained);
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Now the tricky part. In order to manipulate each point, we need to conver the listlist to an
Now the tricky part. In order to manipulate each point, we need to conver the listlist to an array (basically a large matrix). After we evalulate the matrix multiplication ( evalm(...) ) then we convert the result back into a listlist. This is the way I discovered; I hope for the sake of Maple usability that there’s an easier way. For now, just use this.
f2:=convert(evalm(convert(f1, array) &* A), listlist);
f2 :=[ [ 0 0, ], [ 0 10, ], [ -3 10, ], [ -3 8, ], [ -1 8, ], [ -1 6, ], [ -2 6, ], [ -2 4, ], [ -1 4, ], [ -1 0, ],[ 0 0, ]] ...and plotting the resulting set of points gives is what we desired in the first place--the image stretched vertically by a factor of two and flipped across the y-axis.
plot(f2, style=LINE, color=red, thickness=3, view=[-10..10, -10..10], scaling=constrained);
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If we so desired, we could plot both at the same time! We simply set it up as so:
plot({f1, f2}, color=[blue, red], thickness=[1,3], view=[-10..10, -10..10], scaling=constrained);
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Skew image - send (1,0) to (1,1) and (0,1) to (0,1)
A4 := matrix([[1,1],[0,1]]); f4 := convert(evalm(convert(f1, array) &* A4), listlist): plot({f1,f4}, style=LINE, color=[blue,red], thickness=[1,3], view=[-10..10, -10..10], scaling=constrained);
A4 :=
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Rotate Image counter-clockwise 90 degrees - send (1,0) to (0,1) and (0,1) to (-1,0)
A5 := matrix([[0,1], [-1,0]]); f5 := convert(evalm(convert(f1, array) &* A5), listlist): plot({f1,f5}, style=LINE, color=[blue,red], thickness=[1,3], view=[-10..10, -10..10], scaling=constrained);
A5 :=
