I want to compute the orthogonal projection of C^3 onto the image of T(xy,z)=(x-iy+iz,ix-z,2y). I know that im(T)=span{(1,i,,0),(-i,0,2)}. Sor far i have:
#Base estandar de C^3, denotada B
e1 = vector([1, 0, 0]) e2 = vector([0, 1, 0]) e3 = vector([0, 0, 1])
Base de im(T)
b1=vector([1,i,0]) b2=vector([-i,0,2])
Aplicar Gram-Schmidt a b1=u1 y b2 para obtener base ortogonal
u2 = b2 - (b2.dot_product(b1.conjugate()) / b1.dot_product(b1.conjugate())) * b1
Simplificar el vector u2
u2_clean = vector([SR(entry).simplify() for entry in u2])
Mostrar base ortogonal de im(T)
print("Vectores u1 y u2 de la base ortogonal:") show(b1,u2_clean)
Calcular las normas de b1=u1 y de u2
n_u1= sqrt(b1.dot_product(b1.conjugate())) n_u2=sqrt(u2_clean.dot_product(u2_clean.conjugate()))
Construir y mostrar los vectores w1 y w2 de la base ortonormal
w1 = b1/n_u1 w2 = u2/n_u2 print("Vectores w1 y w2 de la base ortonormal:") show(w1,w2)
Definir vector simbólico arbitrario
a,b,c=var("a,b,c") v = vector([a,b,c])
Calcular la proyección ortogonal de C^3 sobre im(T)
P_imT = (v.dot_product(w1.conjugate()))w1 + (v.dot_product(w2.conjugate()))w2 print("P(a,b,c)=") show(P_imT)
Evaluar la proyección en la base estándar
P_e1 = P_imT.subs({a: 1, b: 0, c: 0}) P_e2 = P_imT.subs({a: 0, b: 1, c: 0}) P_e3 = P_imT.subs({a: 0, b: 0, c: 1})
Mostrar los resultados de P(ei)
print("Proyección de e1: ", P_e1) print("Proyección de e2: ", P_e2) print("Proyección de e3: ", P_e3)
Evaluar P(T(ei))
P_Te1 = P_imT.subs({a: 1, b: i, c: 0}) P_Te2 = P_imT.subs({a: -i, b: 0, c: 2}) P_Te3 = P_imT.subs({a: i,b: -1, c: 0}) print("Evaluando P(T(e1)), P(T(e2)), P(T(e3)):") show(P_Te1,P_Te2,P_Te3)
Construir matriz [P]_B a partir de P(ei)
P_B = Matrix([[5/9, -4/9I, -2/9I],[4/9I, 5/9, 2/9],[2/9I, -2/9, 8/9]]) print("[P]_B=") show(P_B)
verificar que la base ortogonal y ortonormal de verdad sean ortogonales.
show(b1.dot_product(u2.conjugate())) show(w1.dot_product(w2.conjugate()))
My concerns here are the I don't know if the formula for P(a,b,c) I found is correct, and subsequently, why the "projection" matrix [P]_B is not idempotent to begin with. I deleted the line in which i computed ([P]_B)^2 but it was not idempotent, so clearly it cannot be an orthogonal projection if it's not a projection to begin with. What did I do wrong here?