# A vector is a list and a matrix is a table

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/3/tee-c3-s0

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c3-s0/

The input [2, 3] is a vector containing two features. A matrix is a rectangle of numbers. “2 by 3” means two rows and three columns. A column vector with two entries has shape 2 by 1. Always write rows before columns.

![Count rows downward and columns across. The shape describes the arrangement of entries.](https://deep-learning-study-room.vercel.app/assets/figures/v_matrix.svg)

Diagram: Count rows downward and columns across. The shape describes the arrangement of entries.. Labels: A vector is a list. A matrix has rows and columns. | A vector is a list. A matrix has rows and columns. | The shape tells you how the values are arranged. | 2 | 1 | 3 | Vector | 3 × 1 | 1 | 2 | 3 | 4 | 5 | 6 | Matrix | 2 rows × 3 columns | Row 1 | Row 2 | A 2 × 3 matrix has six entries. The shape is not the multiplication result.

Count rows downward and columns across. The shape describes the arrangement of entries. Tap the diagram to enlarge.

In this course, unless a question’s diagram explicitly uses another convention, **each row of a weight matrix belongs to one receiving neuron**. Thus W has shape number of output neurons by number of input features, and the input is a column vector.



$$
W=\begin{bmatrix}0.11&0.21\\0.12&0.08\end{bmatrix},\quad x=\begin{bmatrix}2\\3\end{bmatrix},\quad Wx=\begin{bmatrix}0.11(2)+0.21(3)\\0.12(2)+0.08(3)\end{bmatrix}=\begin{bmatrix}0.85\\0.48\end{bmatrix}
$$



**Do it slowly:**  row 1 says 0.11 × 2 = 0.22 and 0.21 × 3 = 0.63, so its answer is 0.85. Row 2 says 0.12 × 2 = 0.24 and 0.08 × 3 = 0.24, so its answer is 0.48. Matrix multiplication is several weighted sums written together.

**Shape check:**  (2 by 2) × (2 by 1) gives (2 by 1). The inside numbers must match; the outside numbers are the answer shape. This check catches many transposed-weight errors before arithmetic begins.

Transpose swaps rows and columns. Elementwise multiplication does not do row-by-column sums: [2, 3] ⊙ [4, 5] = [8, 15]. Dot product does add: [2, 3] · [4, 5] = 8 + 15 = 23. LSTM gates use elementwise multiplication, not a dot product.

**Practice.** Multiply W =  $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$  by x = [2, 1] as a column. Answer: first row 1 × 2 + 2 × 1 = 4; second row 3 × 2 + 4 × 1 = 10. With b = [1, −1], Wx + b = [5, 9].