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# Proof: Matrix determinant gives area of image of unit square under mapping

Proof that the determinant of a 2X2 matrix gives the area of the image of the unit square under the mapping defined by the matrix. Created by Sal Khan.

## Want to join the conversation?

• what it means "unit square under mapping" ? • A mapping is a function, or a transformation. They're all synonyms.

For a given point P in the domain, a function f(x) will send P to a particular point f(P) in the range. That point that it's mapped to is called the image of P, or the image of P under f.

Likewise for whole sets of points. If you have a set S of points in the domain, the set of points they're all mapped to is collectively called the image of S.

If you consider the set of points in a square of side length 1, the image of that set under a linear mapping will be a parallelogram. The title of the video says that if you find the matrix corresponding to that linear transformation, its determinant will be the area of that parallelogram.
• What does dialation mean?
(1 vote) • In common speech, 'dilate' means 'make bigger'. When the pupils of your eyes get larger, we say they are dilated.

In math, 'dilate' means to change the size of something (either bigger or smaller) in a uniform way. If I draw any two lines of the same length on my shape, then dilate the whole figure, the two lines will still be the same length as each other afterward. So if you stretch a square out into a rectangle, that is not a dilation, because it wasn't done uniformly. 