> # hw 2, sections 11.2 and 11.3\
# selected problems\
# load linalg \
with(linalg);
Warning: new definition for   norm
Warning: new definition for   trace


[BlockDiagonal, GramSchmidt, JordanBlock, Wronskian, add, addcol, addrow, adj,

    adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix,

    charmat, charpoly, col, coldim, colspace, colspan, companion, concat, cond,

    copyinto, crossprod, curl, definite, delcols, delrows, det, diag, diverge,

    dotprod, eigenvals, eigenvects, entermatrix, equal, exponential, extend,

    ffgausselim, fibonacci, frobenius, gausselim, gaussjord, genmatrix, grad,

    hadamard, hermite, hessian, hilbert, htranspose, ihermite, indexfunc,

    innerprod, intbasis, inverse, ismith, iszero, jacobian, jordan, kernel,

    laplacian, leastsqrs, linsolve, matrix, minor, minpoly, mulcol, mulrow,

    multiply, norm, normalize, nullspace, orthog, permanent, pivot, potential,

    randmatrix, randvector, rank, ratform, row, rowdim, rowspace, rowspan,

    rref, scalarmul, singularvals, smith, stack, submatrix, subvector,

    sumbasis, swapcol, swaprow, sylvester, toeplitz, trace, transpose,

    vandermonde, vecpotent, vectdim, vector]
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> #11  a+b is add(a,b,1,1), a-b is add(a,b,1,-1), 3a+4b is add(a,b,3,4)\
a :=[5,-12]; b:=[-2,8];

                                  a := [5, -12]

                                  b := [-2, 8]
--------------------------------------------------------------------------------
> add(a,b,1,1);

                                    [ 3, -4 ]
--------------------------------------------------------------------------------
> add(a,b,1,-1);

                                   [ 7, -20 ]
--------------------------------------------------------------------------------
> add(a,b,3,4);

                                    [ 7, -4 ]
--------------------------------------------------------------------------------
> add(a,b,2,0); #or\
scalarmul(a,2);

                                   [ 10, -24 ]

                                   [ 10, -24 ]
--------------------------------------------------------------------------------
> norm(a,2); # this is |a|

                                       13
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> #21\
a:=[-2,4,3];

                                 a := [-2, 4, 3]
--------------------------------------------------------------------------------
> scalarmul(a, 1/norm(a,2));

                               1/2         1/2         1/2
                    [ - 2/29 29   , 4/29 29   , 3/29 29    ]
--------------------------------------------------------------------------------
> evalf(");

                   [ -.3713906763, .7427813528, .5570860147 ]
--------------------------------------------------------------------------------
> #25\
a:='a'; b:='b';# clear the old values of a and b\
eqns:={ a=2*i+3*j,b=i-j };\
solve(eqns,{i,j});

                                     a := a

                                     b := b

                       eqns := {b = i - j, a = 2 i + 3 j}

                    {j = - 2/5 b + 1/5 a, i = 1/5 a + 3/5 b}
--------------------------------------------------------------------------------
> # section 11.3\
# 1\
a:=[2,5];b:=[-3,1]; dotprod(a,b);

                                   a := [2, 5]

                                  b := [-3, 1]

                                       -1
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> #7\
# |a||b|cos(theta)\
2*3*cos(Pi/3);

                                        3
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> #13\
a:=[1,2];b:=[12,-5];arccos(dotprod(a,b)/(norm(a,2)*norm(b,2)));

                                   a := [1, 2]

                                  b := [12, -5]

                                             1/2
                                arccos(2/65 5   )
--------------------------------------------------------------------------------
> evalf("*180/Pi);# answer in degrees

                                   86.05481373
--------------------------------------------------------------------------------
> #27\
a:=[x,1,2];b:=[3,4,x];eqn:=dotprod(a,b)=0;

                                 a := [x, 1, 2]

                                 b := [3, 4, x]

                               eqn := 5 x + 4 = 0
--------------------------------------------------------------------------------
> solve(eqn,x);\


                                      -4/5
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>