In nwfsc-timeseries/atsalibrary: Packages, data and scripts for ATSA course and lab book

library(learnr)
knitr::opts_chunk$set(echo = FALSE)

Topic 1: Building matrices

1. Build a $4 \times 3$ matrix with the number 1 in column 1, number 2 in column 2 and number 3 in column 3.

   r matrix()

matrix(rep(1:3, each=4), 4, 3)

1. Create a matrix with the numbers 1 through 4 in each row (so same values in each column). Assign to the name mat. Extract the elements in the 1st and 2nd rows and 1st and 2nd columns (you'll have a $2 \times 2$ matrix). Show the R code that will do this.

   r mat <- matrix()

mat <- matrix(1:4, 4, 3)
mat[1:2,1:2]

1. Build a $4 \times 3$ matrix with the numbers 1 through 12 by row (meaning the first row will have the numbers 1 through 3 in it).

   r mat <- matrix()

matrix(1:12, 4, 3, byrow = TRUE)

1. Extract the 3rd row of the above. Show R code to do this where you end up with a vector and how to do this where you end up with a $1 \times 3$ matrix.

   r mat <- matrix()

mat <- matrix(1:12, 4, 3, byrow = TRUE)
mat[3, , drop=FALSE]

1. Build a $4 \times 3$ matrix that is all 1s except a 2 in the (2,3) element (2nd row, 3rd column).

   r mat <- matrix()

mat <- matrix(1, 4, 3)
mat[2,3] <- 2
mat

1. Let $\mathbf{A}=\left[ \begin{smallmatrix}1&4&7\2&5&8\3&6&9\end{smallmatrix}\right]$. Build this A matrix in R.

   ```r

   ```

A <- matrix(1:9, 3, 3)
A

Topic 2: Diagonal matrices

quiz(
  question("What is a square matrix?",
    answer("A matrix with the same number of rows and columns.", correct = TRUE),
    answer("A matrix that can always be inverted."),
    answer("A matrix with only integers.")
  ),
  question("A diagonal matrix is always square.",
    answer("TRUE", correct = TRUE),
    answer("FALSE")
  ),
  question("A diagonal matrix has what value on the offdiagonal?",
    answer("0", correct = TRUE),
    answer("1"),
    answer("Can be any value.")
  ),
  question("A diagonal matrix has what value on the diagonal?",
    answer("0"),
    answer("1"),
    answer("Can be any value.", correct = TRUE)
  ),
  question("All values on the diagonal of a diagonal matrix must be the same.",
    answer("TRUE"),
    answer("FALSE", correct = TRUE)
  ),
  question("What is an identity matrix?",
    answer("A square matrix will all 1s."),
    answer("A diagonal matrix with 1s on the diagonal and 0s on the offdiagonal", correct = TRUE),
    answer("A matrix's inverse.")
  ),
  question("If I is a 3 x 3 identity matrix and A is a 3 x 3 matrix, then `I %*% A` equals what?",
    answer("A", correct=TRUE),
    answer("That matrix muliplication is illegal."),
    answer("I")
  ),
  question("If I is a 3 x 3 identity matrix and A is a 3 x 3 matrix, then `A %*% I` equals what?",
    answer("A", correct=TRUE),
    answer("That matrix multiplication is illegal."),
    answer("I")
  ),
  question("If I is a 3 x 3 identity matrix and A is a 3 x 4 matrix, then `A %*% I` equals what?",
    answer("A"),
    answer("That matrix muliplication is illegal.", correct=TRUE),
    answer("I")
  ),
  question("What size must I be for `A %*% I` to be legal? Here I is still an identity matrix.",
    answer("4 x 4", correct=TRUE),
    answer("4 x 3"),
    answer("An identity matrix can never appear on the right in matrix multiplication.")
  )
)

1. Build a $4 \times 4$ diagonal matrix with 1 through 4 on the diagonal.

   ```r

   ```

diag(1:4)

1. Build a $5 \times 5$ identity matrix.

   ```r

   ```

diag(1, 5)

1. Assign the matrix in question 8 to mat and replace the diagonal in mat with 2 (the number 2).

   ```r

   ```

mat <- diag(1, 5)
diag(mat) <- 2
mat

1. Build a $5 \times 5$ matrix with 2 on the diagonal and 1s on the offdiagonals.

   ```r

   ```

mat <- matrix(1, 5, 5)
diag(mat) <- 2
mat

1. Build a $3 \times 3$ matrix with the first 9 letters of the alphabet. First column should be "a", "b", "c". letters[1:9] gives you these letters.

   r mat <-

mat <- matrix(letters[1:9], 3, 3)
mat

1. Replace the diagonal of this matrix with the word "cat".

   ```r

   ```

diag(mat) <- "cat"
mat

1. Create at $3 \times 3$ matrix with 0 (number 0) on the offdiagonals and "cat" on the diagonal. You will need to use list() inside the matrix() call.

   r mat <-

mat <- matrix(list(0), 3, 3)
diag(mat) <- "cat"
mat

quiz(
  question("What happens if you use `diag('cat', 3)`?",
    answer("It works."),
    answer("It puts NA on the diagonal", correct = TRUE),
    answer("It puts the character 0 on the offdiagonal.")
  ),
  question("What happens if you use `mat <- diag(1, 3); diag(mat) <- 'cat'`?",
    answer("It works."),
    answer("It puts NA on the diagonal"),
    answer("It changes number 0 on the offdiagonal to character '0'.", correct = TRUE)
    ),
  question("What happens if you use `mat <- matrix(0, 3, 3); diag(mat) <- 'cat'`?",
    answer("It works."),
    answer("It puts NA on the diagonal"),
    answer("It changes number 0 on the offdiagonal to character '0'.", correct = TRUE)
    )
)

Topic 3. Matrix multiplication

quiz(
  question("`A = B %*% C`. If B is a 2 x 3 matrix, C must have how many rows?",
    answer("1"),
    answer("2"),
    answer("3", correct = TRUE)
  ),
  question("How many columns does C have?",
    answer("2"),
    answer("3"),
    answer("It can have any number of columns.", correct=TRUE)
  ),
  question("`A = B %*% C`. If B is a p x q matrix and C is a q x n matrix, what is the size of A?",
    answer("p x n", correct = TRUE),
    answer("q x n"),
    answer("p x p")
  )
)

1. Build a $4 \times 3$ matrix with all 1s. Multiply this by a $3 \times 4$ matrix with all 2s.

   r mat1 <- mat2 <-

mat1 <- matrix(1, 4, 3)
mat2 <- matrix(2, 3, 4)
mat1 %*% mat2

1. In the equation, $\mathbf{A} \mathbf{B} = \mathbf{C}$, let $\mathbf{A}=\left[ \begin{smallmatrix}1&4&7\2&5&8\3&6&9\end{smallmatrix}\right]$. Build a $\mathbf{B}$ matrix such that $\mathbf{C}=\left[ \begin{smallmatrix}7\8\9\end{smallmatrix}\right]$. Show your R code for $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{A} \mathbf{B}$.

   ```r

   ```

A <- matrix(1:9, 3, 3)
B <- matrix(c(0,0, 1), 3, 1)
A %*% B

1. Let $\mathbf{A}$ be the same as in the previous question. Build a $\mathbf{B}$ matrix such that $\mathbf{B}\mathbf{A}=\left[ \begin{smallmatrix}2 & 5 & 8\end{smallmatrix}\right]$. Show your R code for $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{A} \mathbf{B}$.

   ```r

   ```

A <- matrix(1:9, 3, 3)
B <- matrix(c(0, 1, 0), 1, 3)
B %*% A

1. Make $\mathbf{A}=\left[ \begin{smallmatrix}1&3\2&4\end{smallmatrix}\right]$. Build a $\mathbf{B}$ diagonal matrix such that $\mathbf{A}\mathbf{B}=\left[\begin{smallmatrix}1&6\2&8\end{smallmatrix}\right]$. Show your R code for $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{A} \mathbf{B}$.

   ```r

   ```

A <- matrix(1:4, 2, 2)
B <- diag(1:2)
A %*% B

1. Using the same $\mathbf{A}$ and $\mathbf{B}$ matrices, how would you create $\left[\begin{smallmatrix}1&3\4&8\end{smallmatrix}\right]$ by matrix multiplication?

   ```r

   ```

B %*% A

1. Same $\mathbf{A}$ matrix as above and equation $\mathbf{A} \mathbf{B} = \mathbf{C}$. Build a $\mathbf{B}$ matrix such that $\mathbf{C}=2\mathbf{A}$. So $\mathbf{C}=\left[ \begin{smallmatrix}2&6\ 4&8 \end{smallmatrix}\right]$. Hint, $\mathbf{B}$ is diagonal.

   ```r

   ```

A <- matrix(1:4, 2, 2)
B <- diag(2,2)
A %*% B

Topic 4. Transpose

quiz(
  question("If A is a 4 x 3 matrix, is `A %*% A` possible?",
    answer("No, in matrix multiplication, columns of  the matrix on left must match rows of matrix on right.", correct = TRUE),
    answer("Yes, because a matrix can always be multiplied by itself.")
  ),
  question("What is the size of `t(A)`?",
    answer("4 x 3"),
    answer("3 x 4", correct=TRUE),
    answer("3 x 3"),
    answer("4 x 4")
  ),
  question("If A is a 4 x 3 matrix, is `A %*% t(A)` possible?",
    answer("No, because in matrix multiplication, columns of  the matrix on left must match rows of matrix on right."),
    answer("No, because a matrix can never be multiplied by a transpose of itself."),
    answer("Yes, because now the matrix on right is `t(A)` and that has 3 rows.", correct = TRUE)
  )
)

1. Build $\mathbf{A}$, a $4 \times 3$ matrix with all 1s in the cells. Show how to write $\mathbf{A}\mathbf{A}^\top$ in R. The $\top$ means transpose.

   ```r

   ```

A <- matrix(1, 4, 3)
A %*% t(A)

Topic 5. Column and row sums

1. Make $\mathbf{A}=\left[ \begin{smallmatrix}1&3\2&4\end{smallmatrix}\right]$. Build a $\mathbf{B}$ matrix to compute the row sums of $\mathbf{A}$. So the first row sum would be $1+3$, the sum of all elements in row 1 of $\mathbf{A}$. $\mathbf{C}$ will be $\left[ \begin{smallmatrix}4\ 6\end{smallmatrix}\right]$, the row sums of $\mathbf{A}$. Show $\mathbf{B}$ and the matrix multiplication that makes $\mathbf{C}$.

   ```r

   ```

A <- matrix(1:4, 2, 2)
B <- matrix(1, 2, 1)
C <- A %*% B
C

1. Same $\mathbf{A}$. Build a $\mathbf{B}$ matrix to compute the columns sums of $\mathbf{A}$. So the first column sum would be $1+2$, the sum of all elements in column 1 of $\mathbf{A}$. $\mathbf{C}$ will be $\left[ \begin{smallmatrix}3&7\end{smallmatrix}\right]$, the column sums of $\mathbf{A}$. Show $\mathbf{B}$ and the matrix multiplication that makes $\mathbf{C}$.

   ```r

   ```

A <- matrix(1:4, 2, 2)
B <- matrix(1, 1, 2)
C <- B %*% A
C

nwfsc-timeseries/atsalibrary documentation built on May 11, 2023, 2:25 a.m.