MTH309 Linear Algebra: Maple Guide

Transcription

1 MTH309 Linear Algebra: Maple Guide Section 1. Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) Example. Calculate the REF and RREF of the matrix Solution. Use the following command to insert the matrix entries. Notice the entries in each row are included in brackets [ ]. A:=Matrix([[3,-7,8,-5,8,9],[0,3,-6,6,4,-5],[3,-9,12,-9,6,15]]); (1.1) The following command will display the row echelon form after downward row operations (without row scaling). > GaussianElimination(A); (1.2) Find the Reduced Row echelon form (after down-and-up row operations). > ReducedRowEchelonForm(A); (1.3) Section 4. Matrix Multiplication and inverse Example. Define two matices A= and B = We calculate the expressions and Solution. Use the following command to insert the entries of matrices and. Notice the entries in each row are included in brackets [ ].

2 A:=Matrix([[1,0,1],[0,1,0],[0,1,1]]); B:=Matrix([[1,0],[0,1],[1,1]]); (2.1) The following command calculates the expression > A^2; (2.2) The following command calculates the inverse matrix > A^(-1); (2.3) The following command calculates the expression For matrix multiplication between matrices and we need to put a dot between them. > A.B; (2.4) Example 2. (Multiply matrix with vectors.) Define a matrix a column vector and a row vector We calculate the expressions and Solution. Use the following command to insert the entries of matrix To define the vectors, we use command Vector, and put entires in a pair of brackets. The option [column] and [row] are used when you want to define a column vector or row vector. A:=Matrix([[1,0,1],[0,1,0],[0,1,1]]); v:=vector[column]([1,0,1]) ;

3 u:=vector[row]([1,2,3]); Calculate the expression and Notice the multiplication is denoted by a dot. > u.a; A.v; (2.5) Section 4. Discrete dynamical system, Markov chain Example. Given the initial state vector and transition matrix, calculate the future time sequence. Solution. Use the following command to insert the matrix entries of the transition matrix. Notice the entries in each row are included in brackets [ ]. A:=Matrix([[0.9,0.2],[0.1,0.8]]); (3.1) Insert the vector of initial state as x[0]=. Insert the entries as a row, the command will automatically interprete it as a column. > x[0]:=vector[column]([0.5,0.5]); (3.2) Insert the total number of steps you wish to do, called n. For example, you want perform the iteration 20 times. Set n to be 20 by the following command. > n:=20; (3.3) The following command will perform iteration n times and produce a sequence of vectors x[k] that is the vector after k iterations. To avoid long output, the results are hidden. If you want to display each x[k] in the iteration, change the last colon " : " to a semicolon " ; ". > for i from 1 to n do x[i]:=a.x[i-1]: od : Use the following command to show some particular future state. For example, if you want to see x[15]: > x[15]; (3.4) Plot the vector sequence x[k] in the plane as a point graph vs time k.

4 > with(plots): plot1:=plot({seq([i,x[i][1]],i=0..n)},x=0..n,y=0..1,style= point,color=red): plot2:=plot({seq([i,x[i][2]],i=0..n)},x=0..n,y=0..1,style= point,color=blue): display({plot1,plot2}); Section 10. Inner product Example. Calculate the inner product of vectors and Solution. Use the command Vector([ ]) to insert the vectors. Then use the command DotProduct to calculate the inner product. V1:=Vector([1,-2,1]); V2:=Vector([0,1,3]); DotProduct(V1,V2);

5 1 (4.1) Section 10. orthogonal complement Example. Calculate a basis of the orthogonal complement of the subspace spanned by vectors. Solution. Insert the vectors. V1:=Vector([1,-2,1,0]); V2:=Vector([2,-4,3,1]); V3:=Vector([0,0,1,1]); A:=convert([V1,V2,V3],Matrix);

6 (5.1) Calculate the transpose the matrix A, to form row space > A_Transpose:=Transpose(A); (5.2) Row reduce the tranpose of matrix A to RREF > RREF_of_A_Transpose:=ReducedRowEchelonForm(A_Transpose); (5.3) > Basis_of_Complement:=NullSpace(A_Transpose); (5.4) Section 11. Gram-Schmidt Process Example. Apply the Gram-Schmidt process to vectors,. Solution. Insert the vectors. V1:=Vector[column]([1,-2,1]); V2:=Vector[column]([0,6,6]); V3:=Vector[column]([22,-11,11]);

7 (6.1) The following command perfoms Gram-Schmidt Process and displays the corresponding orthogonal basis; > GramSchmidt([V1,V2,V3]); (6.2) The following command perfoms Gram-Schmidt Process and normalizes the vectors. The outcome is the corresponding orthonormal basis; > GramSchmidt([V1,V2,V3],normalized=true); (6.3) Section 11. QR decomposition Example. Find the QR decomposition of the matrix. Solution. Insert the vectors in a matrix. A:=Matrix([[1,2,0],[2,4,1],[1,2,1],[0,1,0]]); (7.1) The following command will compute the QR decomposition > Q,R:=QRDecomposition(A);

8 (7.2) Section 12. Least Square Problem Example. Find the least squares solution of the system Ax=b for and. Solution. Insert the column vectors in matrix A. > restart:with(linearalgebra): V1:=Vector[column]([1,1,1,1,1]): V2:=Vector[column]([0,1,2,3,4]): A:=convert([V1,V2],Matrix); b:=vector[column]([1.9,3.1,3.9,5.3,5.8]); (8.1) Calculate the matrix A^TA and A^Tb that form the normal equations. > ATA:=Transpose(A).A; ATb:=Transpose(A).b; (8.2)

9 (8.2) Solve the normal equations to obtain the optimal vector x: > x_star:=linearsolve(ata, ATb); numerical_x_star:=evalf[5](x_star); b_star:=evalf[5](a.x_star); Error:=evalf[5](Norm(b-b_star,2)); Define the approximating curve of regression: > g:=numerical_x_star[1]+numerical_x_star[2]*x; (8.3) (8.4) The following commands will plot the data list and the approximating curve in the same chart: > n:=rowdimension(a): plot1:=plot({seq([v2[i],b[i]],i=1..n)},x=v2[1]..v2[n],style= point,color=red,symbol=solidcircle,symbolsize=15): plot2:=plot(g,x=v2[1]..v2[n],color=blue): with(plots):display({plot1,plot2});

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