How determinants determine the scale of the transformation?
Set some context: every linear transformations in R2 can be represented by a 2x2 matrix.
These are linear transformations: basically transformations that map vectors to vectors while preserving scalar addition and multiplication.
Finally, how can we calculate the determinant in the 2x2 case? For any matrix
det(A) = ad * bc
But what does this number represent?
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.
det(A) = ad * bc
But what does this number represent?
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.
Notice that we can separate into three cases
1. Positive determinant (det(A) > 0)
2. Zero determinant (det(A) = 0)
3. Negative determinant (det(A) < 0)
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.
Some Examples:
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
2. 
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.
3. 
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.
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.
What about 3 dimensions?
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.
3. 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.
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.
What about 3 dimensions?
1. Volume instead of area
2. 3 basis vectors
3. What happens when det(A) = 0 now?