G Solution: Representing Data

G.1 Creating vectors and matrices

# Creating matrices
A1 <- matrix(
  c(1:9),
  nrow = 3
)

A2 <- matrix(
  c(1, 3, 4, 6, 9, 2, 1, 0, 3),
  nrow = 3
)

# Creating vectors
v1 <- c(0, 1, 1)
v2 <- c(2, 1, 0)
v3 <- c(3, 1, 1)
x1 <- c(1, 2, 0, 1)
x2 <- c(2, 3, 1, 1)
x3 <- c(4, 1, 2, 0)

# Another way to create matrices
B1 <- cbind(v1, v3, v2)
B2 <- rbind(v1, v2, v3)

G.2 Solution Q2

Q2: Calculate the following matrices.

A1 + A2
A1 * A2
A1 - A2
transpose of A1
transpose of B2

A1 + A2

#>      [,1] [,2] [,3]
#> [1,]    2   10    8
#> [2,]    5   14    8
#> [3,]    7    8   12

A1 * A2

#>      [,1] [,2] [,3]
#> [1,]    1   24    7
#> [2,]    6   45    0
#> [3,]   12   12   27

A1 - A2

#>      [,1] [,2] [,3]
#> [1,]    0   -2    6
#> [2,]   -1   -4    8
#> [3,]   -1    4    6

t(A1)

#>      [,1] [,2] [,3]
#> [1,]    1    2    3
#> [2,]    4    5    6
#> [3,]    7    8    9

t(B2)

#>      v1 v2 v3
#> [1,]  0  2  3
#> [2,]  1  1  1
#> [3,]  1  0  1

G.3 Solution Q3

Q3: Compute the norms of all vectors.

sum(v1^2)

#> [1] 2

sum(v2^2)

#> [1] 5

sum(v3^2)

#> [1] 11

sum(x1^2)

#> [1] 6

sum(x2^2)

#> [1] 15

sum(x3^2)

#> [1] 21

G.4 Solution Q4

Q4: Compute the inverse of A1, A2, and B3. You don’t need to invert the matrices by hand but check they are an inverse.

# Inverse doesn't exist as it is a singular matrix
solve(A1)

#> Error in solve.default(A1): Lapack routine dgesv: system is exactly singular: U[3,3] = 0

solve(A2)

#>            [,1]        [,2]        [,3]
#> [1,] -0.4736842  0.28070175  0.15789474
#> [2,]  0.1578947  0.01754386 -0.05263158
#> [3,]  0.5263158 -0.38596491  0.15789474

solve(A2) %*% A2

#>               [,1]          [,2]          [,3]
#> [1,]  1.000000e+00 -5.551115e-17  2.775558e-17
#> [2,] -2.775558e-17  1.000000e+00 -4.163336e-17
#> [3,] -1.110223e-16  1.665335e-16  1.000000e+00

solve(B2)

#>              v1 v2         v3
#> [1,] -0.3333333  0  0.3333333
#> [2,]  0.6666667  1 -0.6666667
#> [3,]  0.3333333 -1  0.6666667

solve(B2) %*% B2

#>               [,1] [,2] [,3]
#> [1,]  1.000000e+00    0    0
#> [2,]  1.110223e-16    1    0
#> [3,] -1.110223e-16    0    1

G.5 Solution Q5 and Q6

Q5: Compute A1 %*% v1 and v1 %*% A1

A1 %*% v1

#>      [,1]
#> [1,]   11
#> [2,]   13
#> [3,]   15

v1 %*% A1

#>      [,1] [,2] [,3]
#> [1,]    5   11   17

Q6: Compute v2 %*% B2 and B2 %*% v2

v2 %*% B2

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

B2 %*% v2

#>    [,1]
#> v1    1
#> v2    5
#> v3    7

G.6 Solution - Data Processing Function

Here’s a function that extracts a matrix and vector from a data.frame and performs matrix multiplication. Note, this is a more advanced function that includes error handling and checks to ensure the matrix multiplication is valid.Also, this is a implementation and there are several ways to do this:

# Function to extract matrix and vector from data.frame
#   and perform matrix multiplication
# Note: For matrix multiplication A %*% v, the vector length must equal
#   the number of columns in A
# Here we will make use of the function `stop()` to handle errors.
#   this will ensure you know where the error is coming from
matrix_vector_multiply <- function(data, columns, vector, n_rows) {

  # Input validation
  if (!is.data.frame(data)) {
    stop("Error: 'data' must be a data.frame")
  }

  if (!is.numeric(columns) || any(columns < 1) ||
      any(columns > ncol(data))) {
    stop("Error: 'columns' must be valid column indices within the data.frame")
  }

  if (!is.numeric(vector) || length(vector) != 1 || vector < 1 ||
      vector > ncol(data)) {
    stop("Error: 'vector' must be a single valid column index")
  }

  if (!is.numeric(n_rows) || length(n_rows) != 1 || n_rows < 1 ||
      n_rows > nrow(data)) {
    stop("Error: 'n_rows' must be a single positive integer not exceeding ",
         "the number of rows in data")
  }

  # Check if selected columns are numeric
  if (!all(sapply(data[columns], is.numeric))) {
    stop("Error: All selected columns must be numeric")
  }

  if (!is.numeric(data[[vector]])) {
    stop("Error: The vector column must be numeric")
  }

  # Extract the matrix (first n_rows of specified columns)
  matrix_data <- as.matrix(data[1:n_rows, columns])

  # Extract the vector (first ncol(matrix) rows of specified column)
  # For matrix multiplication A %*% v, length(v) must equal ncol(A)
  vector_length_needed <- ncol(matrix_data)

  if (vector_length_needed > nrow(data)) {
    stop("Error: Not enough rows in data to create vector of required ",
         "length (", vector_length_needed, ")")
  }

  vector_data <- as.vector(data[1:vector_length_needed, vector])

  # Print dimensions before multiplication
  cat("Matrix dimensions:", dim(matrix_data), "\n")
  cat("Vector length:", length(vector_data), "\n")

  # Check if matrix multiplication is possible (should always be true now)
  if (ncol(matrix_data) != length(vector_data)) {
    stop("Error: Matrix multiplication not possible - number of columns ",
         "in matrix (", ncol(matrix_data), ") must equal length of vector (",
         length(vector_data), ")")
  }

  # Perform matrix multiplication
  result <- matrix_data %*% vector_data

  # Return the result
  return(result)
}

G.7 Example Usage with Iris Dataset

Let’s demonstrate the function using the iris dataset:

# Load the iris dataset
data(iris)

# Display the structure of iris dataset
str(iris)

#> 'data.frame':    150 obs. of  5 variables:
#>  $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
#>  $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
#>  $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
#>  $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
#>  $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...

# Example 1: Extract first 3 columns as matrix, 4th column as vector,
#   using 5 rows
# Note: Vector will have length 3 (matching matrix columns) for valid
#   multiplication
cat("Example 1: Using columns 1-3 as matrix (5 rows), column 4 as vector",
    "(3 elements)\n")

#> Example 1: Using columns 1-3 as matrix (5 rows), column 4 as vector (3 elements)

result1 <- matrix_vector_multiply(
  data = iris,
  columns = 1:3,
  vector = 4,
  n_rows = 5
)

#> Matrix dimensions: 5 3 
#> Vector length: 3

print(result1)

#>   [,1]
#> 1 2.00
#> 2 1.86
#> 3 1.84
#> 4 1.84
#> 5 2.00

# Example 2: Extract columns 1 and 3 as matrix, column 2 as vector,
#   using 10 rows
# Note: Vector will have length 2 (matching matrix columns) for valid
#   multiplication
cat("Example 2: Using columns 1,3 as matrix (10 rows), column 2 as vector",
    "(2 elements)\n")

#> Example 2: Using columns 1,3 as matrix (10 rows), column 2 as vector (2 elements)

result2 <- matrix_vector_multiply(
  data = iris,
  columns = c(1, 3),
  vector = 2,
  n_rows = 10
)

#> Matrix dimensions: 10 2 
#> Vector length: 2

print(result2)

#>     [,1]
#> 1  22.05
#> 2  21.35
#> 3  20.35
#> 4  20.60
#> 5  21.70
#> 6  24.00
#> 7  20.30
#> 8  22.00
#> 9  19.60
#> 10 21.65

# Example 3: Create a square matrix for demonstration
cat("Example 3: Using columns 1-3 as matrix (3 rows), column 1 as vector",
    "(3 elements)\n")

#> Example 3: Using columns 1-3 as matrix (3 rows), column 1 as vector (3 elements)

result3 <- matrix_vector_multiply(
  data = iris,
  columns = 1:3,
  vector = 1,
  n_rows = 3
)

#> Matrix dimensions: 3 3 
#> Vector length: 3

print(result3)

#>    [,1]
#> 1 49.74
#> 2 46.27
#> 3 45.76

G.7.1 Error Handling Examples

Let’s test the error handling capabilities:

# Test error cases (these will produce errors as expected)

# Error: Invalid column indices
try({
  matrix_vector_multiply(iris, columns = c(1, 10), vector = 2,
                         n_rows = 5)
})

#> Error in matrix_vector_multiply(iris, columns = c(1, 10), vector = 2,  : 
#>   Error: 'columns' must be valid column indices within the data.frame

# Error: Non-numeric vector column (species column)
try({
  matrix_vector_multiply(iris, columns = 1:2, vector = 5,
                         n_rows = 5)
})

#> Error in matrix_vector_multiply(iris, columns = 1:2, vector = 5, n_rows = 5) : 
#>   Error: The vector column must be numeric

# Error: Too many rows requested
try({
  matrix_vector_multiply(iris, columns = 1:2, vector = 3,
                         n_rows = 200)
})

#> Error in matrix_vector_multiply(iris, columns = 1:2, vector = 3, n_rows = 200) : 
#>   Error: 'n_rows' must be a single positive integer not exceeding the number of rows in data

# Error: Non-data.frame input
try({
  matrix_vector_multiply(c(1, 2, 3), columns = 1, vector = 1,
                         n_rows = 1)
})

#> Error in matrix_vector_multiply(c(1, 2, 3), columns = 1, vector = 1, n_rows = 1) : 
#>   Error: 'data' must be a data.frame

G.8 Function Features Summary

The matrix_vector_multiply function includes:

Input Validation:
- Checks if data is a data.frame
- Validates column indices are within bounds
- Ensures vector column index is valid
- Confirms n_rows is appropriate
- Verifies all selected columns are numeric
Smart Dimension Handling:
- Extracts matrix using first n_rows of specified columns
- Extracts vector with length equal to number of matrix columns (ensuring valid multiplication)
- Prints matrix and vector dimensions before multiplication
- Ensures matrix multiplication is mathematically possible
Matrix Multiplication Logic:
- For A %*% v, ensures length(v) == ncol(A)
- Vector is extracted from the first ncol(matrix) rows of the specified column
- This guarantees valid matrix-vector multiplication
Error Handling:
- Provides clear error messages for invalid inputs
- Uses stop() for immediate termination with informative messages
- Checks if enough data rows exist for required vector length
Flexibility:
- Allows selection of any numeric columns for the matrix
- Can extract any number of rows for the matrix (up to dataset size)
- Works with any data.frame containing numeric columns
- Automatically adjusts vector length to match matrix dimensions

F Solution: Confidence Interval

H Solution: Linear regression and Correlation