# 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 A:=Matrix([[1,2,3],[2,-1,1],[3,0,-1]]); b:=Vector([9,8,3]); print("(1)"); A^(-1); print("(2)"); C:=; print("(3)"); R:=linalg[rref](C); print("(3) Alternate"); R:=LinearAlgebra[ReducedRowEchelonForm](C); print("(4)"); R[1..3,4]; # Select col=4 from 3x4 matrix R print("(4) Alternate"); X:=LinearAlgebra[Column](R,4); print("(5)"); A^3; print("(6)"); A^+; print("(7)"); A-3*A^2; print("(8)"); X:=A^(-1).b;X:=LinearAlgebra[LinearSolve](A,b); # (9): linsolve prints nothing: a signal equation "0=3". print("(9)"); F:=Matrix([[1,2,3],[2,-1,1],[0,0,0]]);linalg[linsolve](F,b); ; # Visual explanation of the signal equation print("(10)"); A^+ . A; (A^+.A)^(-1); A^(-1) . (A^(-1))^+; > ># Example 2 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:=linalg[rref](A); # leading ones in columns 1,2,4 BASIScolumnspace=A[1..4,1],A[1..4,2],A[1..4,4]; print("(2)"); F:=linalg[rref](A^+); # leading ones in columns 1,2,3 BASISrowspace=row(A,1),row(A,2),row(A,3); print("(3)"); linalg[nullspace](A); linalg[linsolve](A,Vector([0,0,0,0])); print("(4)"); RANK=linalg[rank](A); NULLITY=linalg[coldim](A)-linalg[rank](A); print("(5)"); DIMnullspace=linalg[coldim](A)-linalg[rank](A); DIMrowspace=linalg[rank](A); DIMcolumnspace=linalg[rank](A); > ># Example 3 A:=Matrix([[ 1, 1, 1, 2, 6], [ 2, 3,-2, 1,-3], [ 3, 5,-5, 1,-8], [ 4, 3, 8, 2, 3]]); v1:=Vector([1,2,3,4]); v2:=Vector([1,3,5,3]); v3:=Vector([2,1,1,2]); w1:=Vector([1,0,0,-3]); w2:=Vector([0,1,0,17]); w3:=Vector([0,0,1,-9]); F:=; G:=; H:=; linalg[rank](F); linalg[rank](G); linalg[rank](H); > ># Example 4 A:=Matrix([[ 1, 1, 1, 2, 6], [ 2, 3,-2, 1,-3], [ 3, 5,-5, 1,-8], [ 4, 3, 8, 2, 3]]); linalg[rref](A); # pivot cols 1,2,4 v1:=A[1..4,1]; v2:=A[1..4,2]; v3:=A[1..4,4];# Name the columns linalg[rref](A^+); # pivot cols 1,2,3 w1:=A[1,1..5]; w2:=A[2,1..5]; w3:=A[3,1..5];# Name the rows ># Example 5 A:=Matrix([[1,2,3],[2,-1,1],[3,0,0]]); B:=A-(-3)*<1,0,0|0,1,0|0,0,1>; linalg[linsolve](B,Vector([0,0,0])); # ans: Vector([-2*t,1*t,2*t]), maple replaces t by _t_1 # Basis == partial on t == Vector([-2,1,2])