Problem set 9

You are not allowed to load any package or use for-loop. For exercises 1 and 3-6 you only get to write one line of code for the solution.

For better preparation for midterm, we recommend not using chatGPT for this homework.

1. Create a 100 by 10 matrix of randomly generated normal numbers. Put the result in x.

2. Apply the three R functions that give you the dimension of x, the number of rows of x, and the number of columns of x, respectively.

3. Add the scalar 1 to row 1, the scalar 2 to row 2, and so on, to the matrix x.

4. Add the scalar 1 to column 1, the scalar 2 to column 2, and so on, to the matrix x. Hint: Use sweep with FUN = "+".

5. Compute the average of each row of x.

6. Compute the average of each column of x.

7. For each digit in the MNIST training data, compute the proportion of pixels that are in a grey area, defined as values between 50 and 205. Make a boxplot by digit class. Hint: Use logical operators and rowMeans.

8. Use the function solve to solve the following system of equations. Hint: use the function solve.

\[ \begin{align} x+2y−2z &=−15\\ 2x+y−5z&=−21\\ x−4y+z&=18 \end{align} \]

9. Use matrix multiplication to compute the average of each column of x and store in a single row matrix. Hint define a \(1\times n\) matrix \((1/n, \dots, 1/n)\) with \(n\) the nrow(x).

10. Use matrix multiplication and other matrix operations to compute the standard deviation of each column. Do not use sweep or apply.