Multiplication and Inverse.sagews

ma = identity_matrix(3)

ma

[1 0 0] [0 1 0] [0 0 1]

ma.swap_rows(0, 1); ma

[0 1 0] [1 0 0] [0 0 1]

test = matrix(QQ, [[1, 2, 3], [4, 5, 6], [7, 8, 9]])

ma*test

[4 5 6] [1 2 3] [7 8 9]

mb = identity_matrix(3); mb.rescale_row(2, -3); mb

[ 1 0 0] [ 0 1 0] [ 0 0 -3]

mb*test

[ 1 2 3] [ 4 5 6] [-21 -24 -27]

mc = identity_matrix(3); mc.add_multiple_of_row(1, 0, 2); mc

[1 0 0] [2 1 0] [0 0 1]

b1 = vector(RR, [1, 1, 0]); b2 = vector(RR, [1, -1, 1]);

b1

(1.00000000000000, 1.00000000000000, 0.000000000000000)

var('s, t, k'); x = 0; y =0; z = 0; var('s, t, k'); solve([s-t+k*(x-6)==3, s+t+k*(x-6)==3, s+t+k*(y+6)==-3, t+k*(z-6) == 3], [s, t, k])

(s, t, k) (s, t, k) [[s == 0, t == 0, k == (-1/2)]]

x = 1; y =0; z = 0; var('s, t, k'); solve([s-t+k*(x-6)==3, s+t+k*(y+6)==-3, t+k*(z-6) == 3], [s, t, k])

(s, t, k) [[s == (6/23), t == (-3/23), k == (-12/23)]]

f1 = vector(RR, [6/23, -3/23])

f1.normalized()

(0.894427190999916, -0.447213595499958)

f1

(0.260869565217391, -0.130434782608696)

x = 0; y =1; z = 0; var('s, t, k'); solve([s-t+k*(x-6)==3, s+t+k*(x-6)==3, s+t+k*(y+6)==-3, t+k*(z-6) == 3], [s, t, k])

(s, t, k) [[s == (6/25), t == (3/25), k == (-12/25)]]

f2 = vector(RR, [6/25, 3/25])

x = 0; y =0; z = 1; var('s, t, k'); solve([s-t+k*(x-6)==3, s+t+k*(y+6)==-3, t+k*(z-6) == 3], [s, t, k])

(s, t, k) [[s == 0, t == (3/11), k == (-6/11)]]

f3 = vector(RR, [0, 3/11])

transM = matrix(RR, [f1.normalized(), f2.normalized(), f3.normalized()]).transpose()

transM

[ 0.894427190999916 0.894427190999916 0.000000000000000] [-0.447213595499958 0.447213595499958 1.00000000000000]

e1 = vector([1, 0, 0])

transM*e1

(0.894427190999916, -0.447213595499958)

veclist1 = [];
for i in range(0, 2):
    for j in [0,1]:
        for k in range(0, 20):
           veclist1.append(vector(RR, [i/20, j, k]))
veclist2 = [];
for i in [0,1]:
    for j in range(0,20):
        for k in [0,1]:
           veclist2.append(vector(RR, [i, j/20, k]))
veclist3 = [];
for i in [0,1]:
    for j in [0,1]:
        for k in range(0,20):
           veclist3.append(vector(RR, [i, j, k/20]))
imglist3 =[]; 
for v in veclist3:
    imglist1.append(point((transM*v).list()));
imglist2 =[]; 
for v in veclist2:
    imglist3.append(point((transM*v).list()));
imglist1 =[]; 
for v in veclist1:
    imglist1.append(point((transM*v).list()));

show(p, xmax=2, xmin=-.5, ymax=2, ymin=-.5)

f3

(0.000000000000000, 0.472377492973330)