FAQ on maple lab 5, Spring 2010
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Problem L5.1. (Matrix Algebra)

(5) Solve for X in BX = v by maple commands rref, linsolve, inverse.
  Coomands are like example (8): 
   rref(augment(B,v); X:=linsolve(B,v); X:=evalm(inverse(B) &* v);

(6) Solve AY = v for Y. Do an answer check using linsolve.
    rref(augment(A,v));  # solve using the last frame algorithm
    X:=linsolve(A,v);

(7) Solve AZ = w. Explain your answer using the three possibilities for
a linear system. Discuss the possible maple reports for (1) no solution
case, (2) unique solution, (3) infinitely many solutions.

    det(A);              # zero, unique solution unlikely
    X:=linsolve(A,v);    # Nothing printed - it means no solution
    rref(augment(A,w));  # signal equation from last row

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Problem L5.2. (Independent Columns)

rref(A);           # pivots in cols 1,2
col(A,1),col(A,2); # [1, 2, 0, 1], [1, 3, 1, 2]
colspace(A);       # {vector([1, 0, -2, -1]), vector([0, 1, 1, 1])}

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Problem L5.3. (Equivalent Bases)

v1:=vector([1,2,0,1]); v2:=vector([1,3,1,2]);
w1:=vector([1, 0, -2, -1]); w2:=vector([0, 1, 1, 1]);
F:=augment(v1,v2);
G:=augment(w1,w2);
H:=augment(v1,v2,w1,w2);
rank(F); rank(G); rank(H);

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Problem L5.4. (Matrix Equations)

with(linalg):
A := matrix([[8, 10, 3], [-3, -5, -3], [-4, -4, 1]]);
T:=matrix([[1,0,0],[0,-2,0],[0,0,5]]);
lambda3:=5; B:=evalm(A-(lambda3)*diag(1,1,1));
linsolve(B,vector([0,0,0]));
# ans3: vector([-_t[1], 0, _t[1]]) = _t[1]*vector([-1,0,1]) 
# Basis3 == partial on t_1 of ans3 == vector([-1,0,1])
Repeat this 2 more times, finding a basis for lambda1=1
and lambda2=-2. Assemble the basis vectors into a matrix P,
following the order lambda1, lambda2, lambda3.
  P:=transpose(matrix([[?,?,?],[?,?,?],[-1,0,1]]));
Check invertibility of P:
  det(P) not zero
Check AP=PT:
  evalm(A&*P-P&*T); # should be the zero matrix