Why are we multiplying Area of P in the statement "Area of L(P) = |Det L| (Area of P)"?

Question

In Introduction to Linear Algebra (2nd edition), Lang gives a proof of the statement

$$\text{Area of $L(P)$} = |\det L| (\text{Area of $P$}).$$

I don't understand he multiplies by "Area of $P$" in this statement.

Why do we need to multiply Area of $P$ on the right-hand side while going from the statement $$\text{Area of $P$} = \lvert\det L\rvert$$ to $$\text{Area of $L(P)$} = \lvert \det L\rvert (\text{Area of $P$})?$$

Please explain in simple terms as I am not very experienced with linear algebra.

Answer 1

He's using the letter $P$ in two different ways. If you read carefully, the argument where he proves that the area of $P$ is $\lvert\det L\rvert$ is based on taking the particular case where $P=L(C)$, where $C$ is the unit square. The other use of $P$ is quite general.

Answer 2

Following is the clear explanation of @TedShifrin's answer using the notion of scaling given by @EthanBolker.

In the statement $$\text{Area of $P$} = \lvert\operatorname{Det}L\rvert,$$ the area of C (which is a unit square, so it has area equal to 1) is scaled to the area of P using the Det L factor.

And in the other statement $$\text{Area of $L(P)$} = \lvert \operatorname{Det} L\rvert (\text{Area of $P$})$$ the area of P is scaled to the area of L(P) using the Det L factor.

(L in the two statements is not the same because they have different associated matrices.)