/1. Create and transform vectors and matrices (the transpose vector (matrix) conjugate transpose of a vector (matrix))./ load("eigen")$; row:read("enter your matrix row:")$; col:read("enter your matrix column :")$; Matrix:entermatrix(row,col); Vector:covect(read("Enter your vector in square bracket []:")); Transpose_Matrix:transpose(Matrix); Transpose_Vector:transpose(Vector);
/2. Generate the matrix into echelon form and find its rank./ row:read("enter your matrix row:")$; col:read("enter your matrix column :")$; Matrix:entermatrix(row,col); Echelon:echelon(Matrix); Rank:rank(Echelon);
/3. Find cofactors, determinant, adjoint and inverse of a matrix./ row:read("enter your matrix row:")$; col:read("enter your matrix column :")$; Matrix:entermatrix(row,col); Cofactors:transpose(adjoint(Matrix)); Determinant:determinant(Matrix); Adjoint:adjoint(Matrix); Inverse:invert(Matrix);
/4. Solve a system of Homogeneous and non-homogeneous equations using Gauss elimination method. Non-homogeneous equation -5x-2y+2z=14 3x+1y-1z=-8 2x+2y-1z=-3/ equation1:read("enter first equation:"); equation2:read("enter second equation:"); equation3:read("enter third equation:"); Matrix:augcoefmatrix([equation1,equation2,equation3],[x,y,z]); Echelon:echelon(Matrix)$; Solution:linsolve([Echelon[1][1]*x+Echelon[1][2]*y+Echelon[1][3]*z=Echelon[1][4],Echelon[2][2]*y+Echelon[2][3]*z=Echelon[2][4], Echelon[3][3]*z=Echelon[1][4]], [x,y,z]);
/Homogeneous equation x+y+7z=0 2x+3y+17z=0 x+2y+z=0/ equation1:read("enter first equation:"); equation2:read("enter second equation:"); equation3:read("enter third equation:"); Matrix:augcoefmatrix([equation1,equation2,equation3],[x,y,z]); Echelon:echelon(Matrix)$; Solution:linsolve([Echelon[1][1]*x+Echelon[1][2]*y+Echelon[1][3]*z=0,Echelon[2][2]*y+Echelon[2][3]*z=0, Echelon[3][3]*z=0], [x,y,z]);
/5. Solve a system of Homogeneous equations using the Gauss Jordan method. 2x+2y+2z=0 x+2y+z=0 3x+y-z=0/ Matrix:matrix([2,2,2,0],[1,2,1,0],[3,1,-1,0]); Echelon:echelon(Matrix); Oper:row(Echelon,1)-(row(Echelon,2)+row(Echelon,3)); Rref:addrow(Oper,row(Echelon,2),row(Echelon,3)); Solution:linsolve([x=0, y=0, z=0], [x,y,z]);
/*6. Generate basis of column space, null space, row space and left null space of a matrix space. */ row:read("enter your matrix row:")$; col:read("enter your matrix column :")$; mat:entermatrix(row,col); ColumnSpace:columnspace(mat); NullSpace:nullspace(mat); RowSpace:columnspace(transpose(mat)); leftNullSpace:nullspace(transpose(mat));
/7. Check the linear dependence of vectors. Generate a linear combination of given vectors of Rn/ matrices of the same size and find the transition matrix of given matrix space. / load("eigen")$; vector1:covect(read("Enter your first vector in square bracket []:"))$; vector2:covect(read("Enter your second vector in square bracket []:"))$; vector3:covect(read("Enter your third vector in square bracket []:"))$; mat:addcol(vector1,vector2,vector3); Echelon:echelon(mat)$; Rank:rank(mat); if Rank=3 then("vector are linear independence")else("vector are linear dependence"); for i: 1 thru 3 do poly:print(mat[i][1]"x"+mat[i][2]"y"+mat[i][3]*"z"=0);
/*8. Find the orthonormal basis of a given vector space using the Gram-Schmidt orthogonalization process. */ load("eigen")$; x: entermatrix (3,4); y: gramschmidt (x); a:y[1][4]; f1:(y[1][1]^2+y[1][2]^2+y[1][3]^2+y[1][4]^2)^0.5$; row1:y[1]/%$; f2:(y[2][1]^2+y[2][2]^2+y[2][3]^2+y[2][4]^2)^0.5$; row2:y[2]/%$; f3:(y[3][1]^2+y[3][2]^2+y[3][3]^2+y[3][4]^2)^0.5$; row3:y[3]/%$; Orthonormal_Basis:addrow(matrix(row1),matrix(row2),matrix(row3));
/*9. Check the diagonalizable property of matrices and find the corresponding eigenvalue and verify the Cayley- Hamilton theorem. */ load("eigen")$; mat:matrix([-4,7,1,4],[6,-16,-3,-9],[12,-27,-4,-15],[-18,43,7,24]); poly:expand(charpoly(mat,x)); Eigenvalues:eigenvalues(mat); Eigenvectors:eivects(mat)[2]; P:determinant(addrow(matrix(Eigenvectors[1][1]),matrix(Eigenvectors[2][1]),matrix(Eigenvectors[2][2]),matrix(Eigenvectors[3][1]))); if P = 0 then("given matrix is not diagonalizable") else ("given matrix is diagonalizable"); is_zero_matrix: is(equal(char_poly_matrix, matrix([0, 0, 0, 0], [0, 0, 0, 0],[0, 0, 0, 0],[0, 0, 0, 0]))); if is_zero_matrix = true then print("It verifies Cayley-Hamilton theorem"); else print("It does not verify Cayley-Hamilton theorem");
/10. Application of Linear algebra: Coding and decoding of messages using nonsingular matrices. then decode it. /
"Enter your message in matrix form (numbers should be equal to alphabetical positions)";
"{A:1,B:2,C:3,D:4,E:5,F:6,G:7,H:8,I:9,J:10,K:11,L:12,M:13,N:14,O:15,P:16,Q:17,R:18,S:19,T:20,U:21,V:22,W:23,X:24,Y:25,Z:26, :27}";
row:read("enter your matrix row:")$;
col:read("enter your matrix column :")$;
h [i, j] :=2/ (i +6 j - 1)$;
mat_de:genmatrix(h,row,col)$;
mat_input:entermatrix(row,col);
decode_message:mat_demat_input;
coded_message:decodeMatrix/mat_de;
/11. Compute Gradient of a scalar field. /
load("vect")$;
F(x,y,z):=read("Enter your Function (like xyexp(z^2)):");
Grad:grad(F(x,y,z));
Express_grad:express(Grad);
Solution:ev(Express_grad,diff);
/12. Compute Divergence of a vector field. /
load("vect")$;
F(x,y,z):=read("Enter your Function (like [x,2y,exp(zy)]):");
Div:div(F(x,y,z));
Express_div:express(Div);
Solution_div:ev(Express_div,diff);
/13. Compute Curl of a vector field./
load("vect")$;
F(x,y,z):=read("Enter your Function (like [x,2y,exp(zy)]):");
Curl:curl(F(x,y,z));
Express_curl:express(Curl);
Solution_curl:ev(Express_curl,diff);