# 1.a
a = sample(-20:20, 100, replace = TRUE)
A = matrix(a, nr = 10, nc = 10)
# 1.b
s = sample(3:8, 1)
b = sample(-20:20, 10*s, replace = TRUE)
B = matrix(b, nr= s, nc= 10 )
# 1.c
c = sample(-20:20, 10*s, replace = TRUE)
C = matrix(c, nr= 10, nc= s)

#2a
B + t(C)
#2b
3*A -5*C %*% B
#2c
Ainv=solve(A)
Ainv %*% A
round(Ainv %*% A)

#3a - pabaigti kitus
det(t(B) %*% B )

#4

D = C%*%t(C)
v = matrix(sample(-20:20, 10), 10, 1)
solve(A, v)
solve(D, v) # det=0, negali spresti

matrix.rank(D)
DD = cbind(D, v) #prijungti stulpeli v
DD
matrix.rank(DD) #R reikalauja kvadratine
z = matrix(sample(0, 11, replace=TRUE), 1, 11) #sukuriama nuline eilute, kad nepakeisti ranko
DDD = rbind(DD, z) #prijungia nuline eilute
matrix.rank(DDD) #isplestines matricos rank ligus 7, originalios 6. rankai nesutampa, reiskia nera sprendimu

#5
E = t(B) %*% B
a = E %*% v
solve(E, a) #nemoka skaiciuoti tokia
matrix.rank(E)
EE = cbind(E,a)
EEE = rbind(EE,z)
matrix.rank(EEE) #rankE = rankEEE, be galo sprendimu

x1 = solveSLE(E,a) #sukurta funkcija
x2 = solveSLE(E,a)
round(E %*% x1)
round(a)

#6
Lambda = eigen(E) # ieskome tikriniu reiksmiu
TR = Lambda$values
TV = Lambda$vectors #vektoriai normuoti - (isvada)
TV1 = TV[1:10, 1]
TV2 = TV[1:10, 10]
norm(TV1, "2") #tikriname norma "2" - saknies is kvadratu sumos, jeigu = 1, reiskia normuoti
norm(TV1, "2")

#vektorius yra tikrinis jeigu tenkinama salyga: Ex=lambda * x
E%*%TV1
TR[1]*TV1

E%*%TV2
TR[10]*TV2
#tikrinti
(E%*%v)/v #nevienodi, tai netikrinis

(E%*%TV2)/TV2 #vienodi, tai tikrinis
#pedsakos lygus tikriniai reiksmiu sumai
matrix.trace(E)
sum(TR)