Question

What is the R code and result in Rstudio

1. The following data gives, for each amount by which an elastic band is stretched over the end of a ruler, the distance that the band moved when released: stretch 46 54 48 50 44 4 52 distance 148 182 173166 109 141 166 Create a data frame in R with two columns that contain "stretch and "dis tance" respectively. Plot the distance versus the stretch using plot) function What trend do you observe? 2. a. In R, define the following matrices -(1 )and b-GP) b. Find the elementwise multiplication of a and b, as well as the matrix multiplication of a and b c. Find matrix c, such that ac b and ca b. d. Explicitly report the eigenvalues and eigenvectors of a and b. 3. What is the difference between a vector and a list in R? What about the difference between a matrix and a dataframe in R? 4. Without using R, provide the output from the following command in R: c = 1 : 16; d = array(c, dim = c(2, 4, 2)); d[1 , 2, 2]

Answer 1

1.

Store the given data into stretch and distance vector.

stretch = c(46,54,48,50,44,42,52)
distance = c(148,182,173,166,109,141,166)

Store distance and stretch vector in a dataframe "df"

df = data.frame(stretch,distance)

You can now type df in R studio to get the contents of dataframe df.

> df
stretch distance
1 46 148
2 54 182
3 48 173
4 50 166
5 44 109
6 42 141
7 52 166

Plot the stretch and distance plot with the below command.

plot(df$stretch,df$distance, pch = 16, xlab = "stretch", ylab = "distance")

The plot shows there is a linear relationship between stretch and distance with exception of one point (Stretch = 44, distance = 109)

2.

a.

Define the matrices a and b with the below commands in R studio.

a = matrix(data = c(2, 1, 1, 2), byrow = TRUE, nrow = 2)
b = matrix(data = c(1, 0, 0, 1), byrow = TRUE, nrow = 2)

b.

Element wise multiplication is done by the command a * b

The output is,

> a * b
[,1] [,2]
[1,] 2 0
[2,] 0 2

Matrix multiplication is done by the command a %*% b

The output is,

> a %*% b
[,1] [,2]
[1,] 2 1
[2,] 1 2

c.

ac = b => c = a^-1b

We can use solve() function in R to get the inverse of a matrix. Run the command c = solve(a) %*% b

The output is shown below.

> c = solve(a) %*% b
> c
[,1] [,2]
[1,] 0.6666667 -0.3333333
[2,] -0.3333333 0.6666667

ca = b => c = ba^-1

Run the command c = b %*% solve(a)

The output is shown below.

> c = b %*% solve(a)
> c
[,1] [,2]
[1,] 0.6666667 -0.3333333
[2,] -0.3333333 0.6666667

We will get the same value of c in both the outputs.

d.

Eigen values and vectors for a matrix is calculated by the eigen() command in R. The output of the command eigen(a) and eigen(b) is shown below.

> eigen(a)
$values
[1] 3 1

$vectors
[,1] [,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068 0.7071068

> eigen(b)
$values
[1] 1 1

$vectors
[,1] [,2]
[1,] 0 -1
[2,] 1 0

Eigen values of a are 3, 1

Eigen vectors of a are (0.7071068, -0.7071068) and (0.7071068, 0.7071068)

Eigen values of b are 1, 1

Eigen vectors of b are (0, -1) and (1, 0)