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Student Name : Sachid Deshmukh

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Problem Set-1

  • Calculate the dot product u.v where u = [0.5, 0.5] and v = [3, ???4]
u = c(0.5, 0.5)
v = c(3, -4)
dot.prod = u %*%  v
print(dot.prod)
##      [,1]
## [1,] -0.5
  • What are the lengths of u and v
length.u = sqrt(u %*% u)
length.v = sqrt(v %*% v)
print(length.u)
##           [,1]
## [1,] 0.7071068
print(length.v)
##      [,1]
## [1,]    5
  • What is the linear combination: 3u ??? 2v?
lin.comb = (3 * u) - (2 * v)
print(lin.comb)
## [1] -4.5  9.5
  • What is the angle between u and v
angle <- function(x,y){
  dot.prod <- x%*%y 
  norm.x <- norm(x,type="2")
  norm.y <- norm(y,type="2")
  theta <- acos(dot.prod / (norm.x * norm.y))
  #convert from radians to degrees
  theta = theta * (180/pi)
  as.numeric(theta)
}

angle(u , v)
## [1] 98.1301

Problem Set-2

substitute_solve <- function(A,B){

p <- nrow(A)
(U.pls <- cbind(A,b))

U.pls[1,] <- U.pls[1,]/U.pls[1,1]

for (i in 2:p){
 for (j in i:p) {
  U.pls[j, ] <- U.pls[j, ] - U.pls[i-1, ] * U.pls[j, i-1]
 }
 U.pls[i,] <- U.pls[i,]/U.pls[i,i]
}

for (i in p:2){
 for (j in i:2-1) {
  U.pls[j, ] <- U.pls[j, ] - U.pls[i, ] * U.pls[j, i]
 }
}
U.pls
}

A <- matrix(c(1,1,3,2,-1,5,-1,-2,4), nrow = 3, ncol = 3, byrow = TRUE)
b <- matrix(c(1,2,6), nrow = 3, ncol = 1)

#Test our function
result = substitute_solve(A,b)
print(result[,4])
## [1] -1.5454545 -0.3181818  0.9545455

Verify Results

solve (A,b)
##            [,1]
## [1,] -1.5454545
## [2,] -0.3181818
## [3,]  0.9545455