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Stretched unit square

Linear algebraEasy

A linear map of the plane sends the vector $(1, 0)$ to $(3, 1)$ and the vector $(0, 1)$ to $(2, 5)$ . By what factor does it multiply the area of every region? Give the factor as a positive integer.

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Hint

The image of the unit square is a parallelogram whose sides are the images of the basis vectors.

Solution

The matrix of the map has the images as its columns: $A = [3125]$ .
Every area is multiplied by $∣ det (A) ∣$ , which is the area of the image of the unit square.
Here $det (A) = 3 \times 5 - 2 \times 1 = 13$ , which is positive, so the map also keeps orientation.
So every area grows by the factor $13$ .

Answer: $13$

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