Problem Set 1

#A
u <- c(.5,.5)
v <- c(3,-4)
u%*%v
##      [,1]
## [1,] -0.5
#B
norm(u, type="2")
## [1] 0.7071068
norm(v, type="2")
## [1] 5
#C
3*u-2*v
## [1] -4.5  9.5
#D
acos(u%*%v/(norm(u, type="2")*norm(v, type="2")))
##          [,1]
## [1,] 1.712693
1.713*180/3.14
## [1] 98.19745

Problem Set 2

M <- matrix(c(1, 1, 3, 1, 2, -1, 5, 2, -1, -2, 4, 6), nrow = 3, byrow = TRUE)
print(M)
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    3    1
## [2,]    2   -1    5    2
## [3,]   -1   -2    4    6
M[2,] <- M[1,] * -2 + M[2,]
M[3,] <- M[1,] + M[3,]
print(M)
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    3    1
## [2,]    0   -3   -1    0
## [3,]    0   -1    7    7
M[c(2,3), ] <- M[c(3,2), ]
print(M)
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    3    1
## [2,]    0   -1    7    7
## [3,]    0   -3   -1    0
M[2,] <- M[2,] * -1
print(M)
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    3    1
## [2,]    0    1   -7   -7
## [3,]    0   -3   -1    0
M[3,] <- M[2,] * 3 + M[3,]
M[1,]<- M[2,] * -1 + M[1,]
print(M)
##      [,1] [,2] [,3] [,4]
## [1,]    1    0   10    8
## [2,]    0    1   -7   -7
## [3,]    0    0  -22  -21
M[3,]<- M[3,] / -22
print(M)
##      [,1] [,2] [,3]       [,4]
## [1,]    1    0   10  8.0000000
## [2,]    0    1   -7 -7.0000000
## [3,]    0    0    1  0.9545455
M[1,]<- M[3,] * -10 + M[1,]
print(M)
##      [,1] [,2] [,3]       [,4]
## [1,]    1    0    0 -1.5454545
## [2,]    0    1   -7 -7.0000000
## [3,]    0    0    1  0.9545455
M[2,]<- M[3,] * 7 + M[2,]
print(M)
##      [,1] [,2] [,3]       [,4]
## [1,]    1    0    0 -1.5454545
## [2,]    0    1    0 -0.3181818
## [3,]    0    0    1  0.9545455