#---------------------------#
#     Loesungen Blatt1      #
#---------------------------#


### Aufgabe 1 ###

v1 = c(1.1,3.3,4.56,-7.77)
v1

v2 = 1:100
v2

v3 = 2*v2
v3

v4 = v3 - 3
v4

v5 = rep(1,100)
v5

v6 = rep(1:4,25)
v6

v7 = seq(500,5,by=-5)
v7

v8 = v2^2
v8

v9 = 1/v8
v9

x = seq(from=0,to=2*pi,length=100)
x



### Aufgabe 2 ###

# 2a)

n = 1:1000
an = (-1)^(n-1) * 1/n
an
sn = cumsum(an)
sn
plot(sn)
limit = rep(log(2),1000)
points(limit,col="red")

# 2b)

n = 1:1000
an = 1/n^2
an
sn = cumsum(an)
sn
plot(sn)
limit = rep(pi^2/6,1000)
points(limit,col="red")



### Aufgabe 3 ###

# 3a)

a1 = seq(2,16,by=2)
a1
A1 = matrix(a1,nrow=2,ncol=4,byrow=TRUE)
A1

a2 = rev(a1)
a2
A2 = matrix(a2,nrow=4,ncol=2)
A2

a1 = c(-2.2,3.3,1.1)
a2 = c(4.56,-7.89,1.23)
a3 = c(800,400,200)
A3 = rbind(a1,a2,a3)
A3

# 3b)

B1 = diag( 100:1 )
B1
head(B1)

B2 = outer( 1:100 , rep(1,100) )
B2
head(B2)

# 3c)

C1 = outer( 1:10 , 1:10 , FUN="+" )
C1 = 1/C1
C1

C2 = outer( 1:10 , 1:10 )
C2 = sin(2*pi*C2/10)
C2



### Aufgabe 4 ###

# 4a)

n = 1000
x = rnorm(n^2)

# 4b)

A = matrix(x,n,n)
A
head(A)

# 4c)

B = t(A) %*% A
head(B)

# 4d) 

res = eigen(B)
names(res)        # just for info
evals = res$values
plot(evals)