How determinants determine the scale of the transformation?

Set some context: every linear transformations in R2 can be represented by a 2x2 matrix. [br][br]These are linear transformations: basically transformations that map vectors to vectors while preserving scalar addition and multiplication. [br][br]Finally, how can we calculate the determinant in the 2x2 case? For any matrix [br][img]data:image/png;base64,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[/img][br]det(A) = ad * bc[br][br][size=85][size=100]But what does this number represent? [br][/size][/size][br]The central idea that we need to understand is that transformations either shrink or expand a space, and determinants represent the how much the transformation 'scales' the original area. [br]
[size=150]Notice that we can separate into three cases[/size][br]1. Positive determinant (det(A) > 0)[br]2. Zero determinant (det(A) = 0)[br]3. Negative determinant (det(A) < 0)[br][br]What is the difference between when det(A) = 4 and det(A) = -4? In both cases, the area of the image doubles but depending on the transformation matrix img(A) will look different. So essentially the area of our image will be scaled by |det(A)| but depending on the sign of the determinant the orientation of our image may be reversed. [br][br]Some Examples:[br]1. When A = the identity matrix, notice that det(A) = 1 and our image size does not change. What about -identity? These are cases when our x and y components are scaled equally[br]2. 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NwIcQD4Pw6ZVvQ6XWZ8AAAAAElFTkSuQmCC[/img]Our now aligns with these new basis vectors notice how the corners of the image now line up. det(A) = 4 and the image has scaled up 4 times the size. [br]3. [img]data:image/png;base64,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[/img]When det(A) = 0 we essentially lose a dimension and our image is squished onto a line. Notice our basis vectors lie on the same line. [br][br]The main difference is that when we have a negative determinant this means that the orientation of the image has changed similar to flipping a piece of paper over. [br][br][size=200]What about 3 dimensions?[/size][br]
[size=150]1. Volume instead of area[br]2. 3 basis vectors[br]3. What happens when det(A) = 0 now?[/size]

Information: How determinants determine the scale of the transformation?