Tutorial 1

STAT394

Contents
1 Introduction 1

2 General Instructions 1

3 Lines 1
3.1 Restrictions on lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Typing matricies 4
4.1 A 2 × 2 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2 Transpose this matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.3 A 3 × 3 diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.4 Printing nice matricies from R . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Operations with matrices 7

1 Introduction
In this tutorial we construct plots of lines and with restrictions. The tutorial will also cover
typing matrices and working with matrices in R. The amsmath package is used for typing
matricies.

2 General Instructions
Only load those R packages that you effectively use. This will help check if you are reusing
somebody else’s code.
Always use relative paths. You will share your code, and somebody will run it on a different
computer.

3 Lines
A line in R2 is the set of points that satisfy y = β0 + β1 x, with β0 , β1 ∈ R.

1
Figure 1 is my first example. This example was produced by defining the starting and end
points of a segment. Can you think of other ways of defining a straight line, for instance by
stipulating the real numbers β0 and β1 ? How about a point and an angle?
ggplot() +
# geom_hline(yintercept=0) +
# geom_vline(xintercept=0) +
geom_segment(aes(x=-1,
y=1,
xend=1,
yend=-1),
linewidth=2, col="red", alpha=.3) +
geom_point(aes(x=-1, y=1), size=2) +
geom_point(aes(x=1, y=-1), size=2) +
xlim(c(-1.5, 1.5)) +
ylim(c(-1.5, 1.5)) +
coord_fixed() +
labs(x=expression(italic(x)),
y=expression(italic(y))) +
theme(text=element_text(size=12,
family="serif"))

Figure 2 plots the intersection of two lines without the use of the ggplot library.
plot(x=c(-100, 100), y=c(-100, 100), xlab="x",
ylab="y", ylim=c(-10, 10), xlim=c(-10, 10),
main="Intersection of y=x and y=2-x")
grid(nx = NULL, ny = NULL, lty = 2, col = "gray", lwd = 1)
abline(h=0)
abline(v=0)
abline(a=0, b=tan(45*pi/180), col="red", lwd=2)
abline(a=2, b=tan(-pi/4), col="red", lwd=2)
points(x=1, y=1, col="blue", cex=0.8)
text(x=8, y=6, label="y=x")
text(x=-6, y=6, label="y=2-x")

3.1 Restrictions on lines

In the lectures we discussed placing restrictions on the lines. In Figure 3 we have drawn
y = 4x + 2 for x ∈ [2, 4].
ggplot(data = data.frame(x=c(-1, 10), y=c(-1, 20)), aes(x=x, y=y)) +
geom_point(aes(x, y), size=0) +
geom_hline(aes(yintercept = 0)) +
geom_vline(aes(xintercept = 0)) +
geom_abline(aes(slope=4, intercept = 2), color="black", linewidth=.2) +

2
 1

0
y

−1

−1 0 1
x

Figure 1: My very first straight line, with several improvements over a plain plot

3
 Intersection of y=x and y=2−x

10

y=2−x y=x
5
0
y

−5
−10

−10 −5 0 5 10

Figure 2: Intersection of two lines

geom_segment(x=2, xend=4, y=10, yend=4*4+2, color="red", linewidth=2) +

coord_fixed() +
labs(title=expression(paste(italic(y) == 4 ~ italic(x) + 2,
" such that ",
2<= ~ italic(x) <= 4
)
),
xlab=expression(italic(x)),
ylab=expression(italic(y))) +
theme_tufte() +
theme(plot.margin = unit(c(0,0,0,0), "lines"),
title =element_text(size=8))

4 Typing matricies
4.1 A 2 × 2 matrix
$$
A = \begin{bmatrix}
4 & 2 \\

4
 y = 4 x + 2 such that 2 ≤ x ≤ 4

20

15

10
y

0.0 2.5 5.0 7.5 10.0

x

Figure 3: My very first straight line, with several improvements over a plain plot

5
1 & -2
\end{bmatrix}
$$

Produces: " #
4 2
A=
1 −2

4.2 Transpose this matrix

$$
Aˆ{T} = \begin{bmatrix}
4 & 1 \\
2 & -2
\end{bmatrix}
$$

Produces: " #
4 1
AT =
2 −2

4.3 A 3 × 3 diagonal matrix

$$
\begin{bmatrix}
1 & 0 & 0\\
0 & -3 & 0 \\
0 & 0 & 2 \\
\end{bmatrix}
$$

Produces:  
1 0 0
0 −3 0
 

0 0 2

4.4 Printing nice matricies from R

This function prints nicely formatted matricies.
# Writes an R matrix as a matrix
# From https://stackoverflow.com/questions/45591286/for-r-markdown-how-do-i-display-a-m
write_matex2 <- function(x) {
begin <- "\\begin{bmatrix}"
end <- "\\end{bmatrix}"
X <-

6
 apply(x, 1, function(x) {
paste(
paste(x, collapse = "&"),
"\\\\"
)
})
paste(c(begin, X, end), collapse = "")
}

As an example:
A<-matrix(c(1, 0, 0, 0, 2, 0, 0, 0, 1), nrow=3, ncol=3)

In text we write:
$$
A=`r write_matex2(A)`.
$$

This produces:  
1 0 0
A = 0 2 0 .
 

0 0 1

5 Operations with matrices

(a <- c(1, 2, 3)) # A vector

## [1] 1 2 3
(b <- c(3, 4, 5)) # A vector

## [1] 3 4 5
dim(a) # Vectors don't have "dim", they have "length"

## NULL
length(a)

## [1] 3
dim(t(t(a))) # But they receive "dim" once transformed

## [1] 3 1
# By default, they are column vectors

7
a <- matrix(a, nrow=length(a), ncol=1) # to be on the safe side
dim(a)

## [1] 3 1
b <- matrix(b, nrow=1, ncol=length(b))
dim(b)

## [1] 1 3
b %*% a # Inner product of compatible matrices

## [,1]
## [1,] 26
(M <- matrix(c(2, 3, 5, 1, -8, 4), nrow=2))

## [,1] [,2] [,3]

## [1,] 2 5 -8
## [2,] 3 1 4
dim(M)

## [1] 2 3
3 * M # Scalar times a matrix

## [,1] [,2] [,3]

## [1,] 6 15 -24
## [2,] 9 3 12
(M1 <- matrix(c(1, 2, 5, 3, 5, 9, 7, 8, 7), nrow = 3))

## [,1] [,2] [,3]

## [1,] 1 3 7
## [2,] 2 5 8
## [3,] 5 9 7
# Colum-wise creation of a matrix

(M2 <- matrix(c(1, 2, 5, 3, 6, 9, 7, 8, 7), nrow = 3, byrow = TRUE))

## [,1] [,2] [,3]

## [1,] 1 2 5
## [2,] 3 6 9
## [3,] 7 8 7
# Row-wise creation of a matrix

M1 %*% M2

8
## [,1] [,2] [,3]
## [1,] 59 76 81
## [2,] 73 98 111
## [3,] 81 120 155
# Determinants
det(M1)

## [1] -8
det(M2)

## [1] -36
# Inverses
solve(M1)

## [,1] [,2] [,3]

## [1,] 4.625 -5.25 1.375
## [2,] -3.250 3.50 -0.750
## [3,] 0.875 -0.75 0.125
solve(M2)

## [,1] [,2] [,3]

## [1,] 0.8333333 -0.7222222 3.333333e-01
## [2,] -1.1666667 0.7777778 -1.666667e-01
## [3,] 0.5000000 -0.1666667 -3.965082e-18
eigen(M1)

## eigen() decomposition
## $values
## [1] 16.8815273 -4.0000000 0.1184727
##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.4211382 -0.6396021 -0.8180022
## [2,] -0.5542791 -0.4264014 0.5587920
## [3,] -0.7179257 0.6396021 -0.1364692