# Problem L5.2 >A:=matrix([[ 1, 1, 1, 2, 6], > [ 2, 3,-2, 1, -3], > [ 0, 1,-4,-3,-15], > [ 1, 2,-3,-1, -9]]); # Problem L5.3 > A := matrix([[8, 10, 3], [-3, -5, -3], [-4, -4, 1]]); > T:=matrix([[1,0,0],[0,-2,0],[0,0,5]]); ># Example 1 > with(linalg): > A:=matrix([[1,2,3],[2,-1,1],[3,0,-1]]); > b:=vector([9,8,3]); > print("(1)"); inverse(A); > print("(2)"); C:=augment(A,b); > print("(3)"); R:=rref(C); > print("(4)"); X:=col(R,4); > print("(5)"); evalm(A^3); > print("(6)"); transpose(A); > print("(7)"); evalm(A-3*(A^2)); > print("(8)"); X:=linsolve(A,b); X:=evalm(inverse(A) &* b); > print("(9)"); F:=matrix([[1,2,3],[2,-1,1],[0,0,0]]);linsolve(F,b); > # Nothing is printed, because of a signal equation "0=3". > print("(10)"); evalm(transpose(A) &* A); > evalm(inverse(transpose(A) &* A)); > evalm(inverse(A)&*transpose(inverse(A))); > ># Example 2 > with(linalg): > A:=matrix([[ 1, 1, 1, 2, 6], > [ 2, 3,-2, 1,-3], > [ 3, 5,-5, 1,-8], > [ 4, 3, 8, 2, 3]]); > print("(1)"); C:=rref(A); # leading ones in columns 1,2,4 > BASIScolumnspace=col(A,1),col(A,2),col(A,4); > print("(2)"); F:=rref(transpose(A)); # leading ones in columns 1,2,3 > BASISrowspace=row(A,1),row(A,2),row(A,3); > print("(3)"); nullspace(A); linsolve(A,vector([0,0,0,0])); > print("(4)"); RANK=rank(A); NULLITY=coldim(A)-rank(A); > print("(5)"); DIMnullspace=coldim(A)-rank(A); DIMrowspace=rank(A); > DIMcolumnspace=rank(A); > ># Example 3 > with(linalg): > A:=matrix([[ 1, 1, 1, 2, 6], > [ 2, 3,-2, 1,-3], > [ 3, 5,-5, 1,-8], > [ 4, 3, 8, 2, 3]]); > v1:=col(A,1); v2:=col(A,2); v3:=col(A,4); > w1:=vector([1, 0, 0, -3]); w2:=vector([0, 1, 0, 17]); > w3:=vector([0, 0, 1, -9]); > F:=augment(v1,v2,v3); > G:=augment(w1,w2,w3); > H:=augment(F,G); > rank(F); rank(G); rank(H); > > B:=rref(transpose(A)); # pivot cols 1,2,3 > w1:=row(B,1); w2:=row(B,2); w3:=row(B,3); > ># Example 4 > with(linalg): > A:=matrix([[1,2,3],[2,-1,1],[3,0,0]]); > linsolve(evalm(A-(-3)*diag(1,1,1)),vector([0,0,0])); > # ans: t_1*vector([-2,1,2]) > # Basis == partial on t_1 == vector([-2,1,2]) >