{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 0 "" }{TEXT 256 38 "Illustrat ion of 3 by 3 Linear Systems " }}{PARA 19 "" 0 "" {TEXT -1 13 "J. A. Z iegler" }}{PARA 256 "" 0 "" {TEXT -1 55 "Web Page: http://www2.SPSU.e du/math/ziegler/index.html" }}{PARA 256 "" 0 "" {TEXT -1 26 "E-mail: \+ jziegler@spsu.edu" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "October 2, 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 623 "By the analytical aspect of a mat hematical question, we mean its statement in terms of symbols, often r epresenting real numbers, and its resolution by means of the allowable manipulation of those symbols. By its geometric aspect, we naturally mean its expression in terms of geometric objects, lines, surfaces, a nd so on, and their relationships to each other. The interplay of the se two points of view is often very useful in finding the answer to th e question and is especially valuable in understanding both. The rela tively easy creation of the geometrical objects with Maple is an impor tant step forward in our time." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 310 "As an example of this important i dea, we illustrate that much dreaded topic from elementary algebra, th e solution of three linear equations in three unknowns. Because we wa nt to concentrate on the geometric aspect, which is hard to illustrate by hand, we will let Maple do the algebra as well as the pictures." } }}{EXCHG {PARA 3 "" 0 "" {TEXT -1 66 "What Sort of Graph Does a Linear Equation in Three Variables Have?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots):with(plottools):with(linalg):" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "First, we define the equation. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eq1:=x+y+z=1;" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Solving for z, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "We plot as points a number of sol utions of this equation and display them together. Note: So that the \+ surface containing the solutions is easier to see, the " }{TEXT 257 4 "grid" }{TEXT -1 129 " option has been used to increase the number of \+ points used in our 20 x 20 plotting region from the default 25 x 25 to 40 x 40. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "G:=plot3d(z1 ,x=-10..10,y=-10..10,grid=[40,40],color=blue):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "display(G,scaling=constrained,axes=normal,styl e=point);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 294 "Click on the picture. Holding down the \+ left mouse button, move the mouse. Clearly, the graph is a plane. It may be shown that this is always the case for a linear equation in th ree variables. Using the mouse, we may change the style of the displa y so that the graph may be seen more easily." }}}{EXCHG {PARA 3 "" 0 " " {TEXT -1 74 "A System with an Unique Solution: Three Planes Which I ntersect at a Point" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Now consider the system" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:=x+y+z=1;eq2:=x+y-z=1;eq 3:=-2*x+3*y-z=4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Does this sys tem have a solution?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sol ve(\{eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Now consider the determinant of the matrix of coeffiecients for the s ystem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "M:=matrix(3,3,[1, 1,1,1,1,-1,-2,3,-1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det (M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "That this is not zero ma y be shown to be true only of systems which have an unique solution. \+ This determinant is therefore a good diagnostic tool and worth calcula ting." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Because we have three linear equations, e ach represent a plane. The unique solution, ( " }{XPPEDIT 18 0 "-1/5 ,6/5,0;" "6%,$*&\"\"\"\"\"\"\"\"&!\"\"F(*&\"\"'F&\"\"&F(\"\"!" }{TEXT -1 190 " ), is the only point that lies on all three planes. To see \+ this and the relationship between the planes, we graph them and use a \+ small sphere of a contrasting color to indicate the point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z2:=solve(eq2,z);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "z3:=solve(eq3,z);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "a:=plot3d(z1,x=-4..4,y=-2..4,color=yellow,scaling=c onstrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "b:=plot3d(z 2,x=-4..4,y=-2..4,color=red,scaling=constrained):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "c:=plot3d(z3,x=-4..4,y=-2..4,color=green,sca ling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "S:=sp here([-1/5,6/5,0],.4,color=blue,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "display(a,b,c,S,view=[-4..4,-2..4,-5..5], style=patchcontour,scaling=constrained,axes=box);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 65 "Syst ems with No Solution, Case 1: All of the Planes are Parallel" }}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "eq1:=x+y+z=-6; eq2:=x+y+z=1; eq3:=x+y+z=8;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1 ,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "No outp ut has been produced, which means that Maple cannot find a solution. \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Calculating our determinant, \+ we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "M:=matrix(3,3,[1 ,1,1,1,1,1,1,1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "This is a quite different r esult than when we had an unique solution." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Again, \+ we graph the planes defined by the equations of our system. Solving e ach of our equations for z, we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z2:=solve(eq2,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z3:=solve(eq3,z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Plotting and displaying these, we see why there is no solution . Clearly, there is no point which lies on all three planes." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A:=plot3d(z1,x=-10..10,y=-10 ..10,color=yellow,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "B:=plot3d(z2,x=-10..10,y=-10..10,color=red,scaling=co nstrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "C:=plot3d(z3 ,x=-10..10,y=-10..10,color=green,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "display(A,B,C,view=[-10..10,-10..10 ,-10..15],style=patchcontour,scaling=constrained,axes=box);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 66 "A System with No Solution, Case 2: Two of the Planes are Parallel" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:=x+y+z=-3;eq2:=x+y+z=5;e q3:=x+y-5*z=-15;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve( \{eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "No output has been produced, which means that Maple cannot find a soluti on. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "And the determinant?" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "M:=matrix(3,3,[1,1,1,1,1,1, 1,1,-5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Again, zero." }}}{EXCHG {PARA 4 " " 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 " We graph the planes defined by the equations of our system. Solving e ach of our equations for z, we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z2:=solve(eq2,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z3:=solve(eq3,z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Plotting and displaying these, we see why there is no solution . There is no point which lies on all three planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A:=plot3d(z1,x=-10..10,y=-10..10,color=ye llow,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "B:=plot3d(z2,x=-10..10,y=-10..10,color=red,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "C:=plot3d(z3,x=-10..10,y= -10..10,color=green,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "display(A,B,C,view=[-10..10,-10..10,-10..15],style= patchcontour,scaling=constrained,axes=box);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 62 "A System w ith No Solution, Case 3: No Two Planes are Parallel" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:=x+y+z=-3;eq2:=x+y-z=1;eq3:=x+y-5*z=-25;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\},\{x,y, z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "No output has been produ ced, which means that Maple cannot find a solution. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The determinant?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M:=matrix(3,3,[1,1,1,1,1,-1,1,1,-5]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 235 "Geometrically, we can see that we have investigat ed examples of the only three possible cases in which no solution exis ts. It seems as though if there is no solution, then the determinant \+ will be zero. This can be shown to be true. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We graph the planes defined by the equations of our system. Solving eac h of our equations for z, we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z2:=solve(eq2,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z3:=solve(eq3,z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Plotting and displaying these, we see why there is no solution . There is no point which lies on all three planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A:=plot3d(z1,x=-10..10,y=-10..10,color=ye llow,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "B:=plot3d(z2,x=-10..10,y=-10..10,color=red,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "C:=plot3d(z3,x=-10..10,y= -10..10,color=green,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "display(A,B,C,view=[-10..10,-10..10,-10..15],style= patchcontour,scaling=constrained,axes=box);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 45 "A System w ith an Infinite Number of Solutions" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq1:=x +y+z=1;eq2:=-x+z=0;eq3:=x+2*y+3*z=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This shows that there is a solution for each valu e of z: hence, an infinite number of solutions. " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 27 "What about the determinant?" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "M4:=matrix(3,3,[1,1,1,-1,0,1,1,2,3]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(M4);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 199 "So it is not true that if the determinant is zero , then no solution exists. This is a nice example of the importance o f not making a common error in logic: confusing a statement with its c onverse. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Following our usual prodedure, we \+ may see the geometrical relationships between the planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z2:=solve(eq2,z);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "z3:=solve(eq3,z);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "A:=plot3d(z1,x=-10..10,y=-10..10,color=yellow,scali ng=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "B:=plot 3d(z2,x=-10..10,y=-10..10,color=red,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "C:=plot3d(z3,x=-10..10,y=-10..10,co lor=green,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "display(A,B,C,view=[-10..10,-10..10,-10..15],style=patchcontou r,scaling=constrained,axes=box);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "If we like, we can use our solutions to graph the line of intersec tion of the planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "L:=s pacecurve([z,-2*z+1,z],z=-5..6,color=blue,thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "display(A,B,C,L,view=[-10..10,-10.. 10,-10..15],style=patchcontour,scaling=constrained,axes=box);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 308 "An explanation of how the line is described mathematically requires the use of vectors in three dimensi onal space and so is far beyond the level of the usual introductory al gebra course. However, it might provide a useful starting point for t he independent study of vectors by an unusually talented student." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 13 "A Last Remark" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1007 "I n our discussion, we have assumed that the equations are of distinct p lanes. If this is not so, a few so-called degenerate cases may arise. These are easiest to describe geometrically. In the first, there is really only one plane and the equations are then non-zero multiples o f each other. If so, the coordinates of every point in the plane is a solution. Second, there may be only two planes. It is geometrically \+ obvious that these either intersect along a line or are parallel. Fro m the previous case, it is clear that two of the equations must be no n-zero multiples of each other, but not of the third. Now suppose tha t this third equation is ax + by + cz = d. Only if there is a non-zer o number, k, so that one of the other equations can be written as kax \+ + kby + kcz = d1 (the value of d1 is of no interest), are the planes p arallel. In this case, there are obviously no solutions. Otherwise, \+ the planes intersect along a line and the coordinates of each point on this line are solutions. " }}}}{MARK "4 0 0" 41 }{VIEWOPTS 1 1 0 1 1 1803 }