Matrix Operations

Dotting weights with one flat gives one prediction. The agency has thousands of flats, and a loop over them is slow and clumsy. One matrix multiplication predicts every rent at once, and nearly every "shape mismatch" error in ML code comes from not knowing when that multiplication is allowed.

The rule

Inner sizes must match; outer sizes give the answer's shape.

Take the flats as a matrix $X$ with 5 rows (flats) and 3 columns (size, rooms, metro km), and the weights as a column $w$ with 3 rows: [0.3, 1, -1].

• Addition works entry by entry, so both matrices need the same shape.
• Scalar multiplication multiplies every entry: $2X$ doubles every number.
• Transpose swaps rows and columns: $X^T$ is 3 × 5.
• Matrix multiplication $AB$ needs the columns of $A$ to equal the rows of $B$. Each entry of the result is a row of $A$ dotted with a column of $B$.

Writing shapes side by side, $(5 \times 3)(3 \times 1)$: the inner 3s match, so $Xw$ exists, and the outer sizes give a 5 × 1 result, one prediction per flat.

In NumPy, @ is matrix multiplication, while * multiplies entry by entry, stretching shapes where it can. So X * w runs without any error but returns a 5 × 3 table of products, not five predictions.

Order matters. $AB$ and $BA$ are usually different, and often only one of them exists at all.

The shortcut

Write the shapes next to each other and cancel the middle pair. If the middle pair differs, stop: the product is undefined. If it matches, what remains on the outside is the shape of the answer.

The same trick checks $X^T X$, which appears in the formula for linear-regression weights: $(3 \times 5)(5 \times 3)$ cancels the 5s and gives 3 × 3, one row and column per feature.

Drills

Answer fast — trust the rule.

Remember

• Inner sizes must match; outer sizes give the answer's shape.
• Shortcut: write shapes side by side and cancel the middle pair.
• Trap: assuming AB equals BA, or confusing entry-by-entry with matrix multiplication.