Lesson 2 of 3

The determinant.

About 20 minutes · 5 practice questions

Read the full article: Vectors and transformations

Every linear map scales every area by the same factor, the determinant ad − bc: why turns and shears keep area, what a negative determinant says, and what happens when it is zero.

Step 1 of 5 · The area factor

Step 1 of 5

The area factor

A linear map takes the unit square to a parallelogram with sides $M\hat\imath = (a, c)$ ⁠ and $M\hat\jmath = (b, d)$ ⁠. The area of that parallelogram is the absolute value of the number

$\det M = ad - bc$ ⁠

the determinant of $M$ ⁠. And since any shape can be filled with tiny squares, each of which grows in the same proportion, every area in the plane is multiplied by $|\det M|$ ⁠. Here $M$ ⁠ is built from the Article’s stretch, shear and turn. The card shows $\det M$ ⁠ as you drag Apply the transformation; before the map it is $1$ ⁠.

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Step 2 of 5

Every shape

The F is not a square, but its area is multiplied by the same factor too. Work out its new area.

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Step 3 of 5

A negative determinant

$\det M$ ⁠ can be negative. Its size is still the area factor, and the sign says the map has flipped the plane over: what went anticlockwise now goes clockwise. Here Stretch is $-1$ ⁠. Predict what will happen to the F, then watch.

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Step 4 of 5

A zero determinant

When $\det M = 0$ ⁠ the parallelogram is flat: $M\hat\imath$ ⁠ and $M\hat\jmath$ ⁠ lie on one line, and the whole plane is squashed onto a line or a point. Such a map cannot be undone, because many points land in the same place. Find a setting of the controls that squashes the plane.

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Step 5 of 5

What changes area

A turn moves the plane as a rigid body, and a shear slides rows without changing their height. Try each control on its own, watch $\det M$ ⁠ on the card, and answer.

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Still of the Vectors and transformations 3D scene

Interactive scene

Loading the interactive version…

Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Calculate

$M$ ⁠ is built from a stretch of $1.5$ ⁠, a shear of $0.5$ ⁠ and a turn of $30^\circ$ ⁠. By what factor will every area in the plane grow? Answer, then drag Apply the transformation and read $\det M$ ⁠ on the card.

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Practice · Calculate

Before the transformation the F’s area is about $1.055$ ⁠. What will its area be after the same $M$ ⁠?

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Practice · Predict, then look

Stretch is $-1$ ⁠, with no shear and no turn: $\hat\imath$ ⁠ goes to $(-1, 0)$ ⁠ and $\hat\jmath$ ⁠ stays put. What will happen to the F?

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Practice · Reach the target

Squash the whole plane onto one line: change the controls so that $\det M = 0$ ⁠, then press Check.

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Practice · Choose

Which of the controls change the F’s area? Select all that do. Try each on its own.