Checkpoint

Matrices are machines

Twelve questions on matrices as maps, products and shapes, with new numbers and a few classic traps. Work them without the lessons open. You need 80% to pass. The code question needs one line of numpy.

1
A linear map sends $(1, 0)$ to $(3, -1)$ and $(0, 1)$ to $(2, 5)$ . Which matrix represents it?
2
Compute $\mathbf{A}\mathbf{x}$ for $\mathbf{A} = \begin{bmatrix} 1 & 0 & -2 \\ 3 & 1 & 1 \end{bmatrix}$ and $\mathbf{x} = (1, 4, -1)$ .
Ax
3
Which of these can be done by multiplying by a $2 \times 2$ matrix? Select all that apply.
4
Find the single matrix that first applies the shear $\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and then stretches horizontally by 3 with $\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}$ .
5
Let $\mathbf{A} = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 3 & 4 \end{bmatrix}$ and $\mathbf{B} = \begin{bmatrix} 1 & 2 \\ 0 & -1 \\ 5 & 1 \end{bmatrix}$ . What is $(\mathbf{AB})_{21}$ , the entry in row 2 and column 1?
6
In numpy, a has shape (8, 1) and b has shape (3,). What does a + b do?
7
An MLP maps 20 input features through two hidden layers of 50 units each to 5 outputs. Every layer is dense with a bias. How many parameters does it have?
8
Which expression equals $(\mathbf{ABC})^\top$ for matrices whose shapes fit?
9
$\mathbf{A}$ is $100 \times 200$ , $\mathbf{B}$ is $200 \times 300$ and $\mathbf{C}$ is $300 \times 10$ . You compute $\mathbf{ABC}$ with whichever grouping is cheaper. How many multiply-adds does it take?
10
Two dense layers with no nonlinearity between them compute $(\mathbf{X}\mathbf{W}_1 + \mathbf{b}_1)\mathbf{W}_2 + \mathbf{b}_2$ in the row-batch convention. Which single layer $\mathbf{X}\mathbf{W} + \mathbf{b}$ gives the same outputs?
11
Write the matrix that rotates the plane a quarter turn clockwise.
12
Code it
A colleague stores a layer's weights in the math convention: W has shape (d_out, d_in), so one input column becomes $\mathbf{W}\mathbf{x} + \mathbf{b}$ . Your data is a batch X of shape (batch, d_in), one example per row. Write layer(X, W, b) so that it returns every example's output as a row, shape (batch, d_out), without a Python loop.
⌘↵ runsPython sleeps until you run code

12 still unanswered. Unanswered questions count as wrong.