Determinant of Transformation
It’s quite easy to calculate, and not too hard to understand what’s behind it.
The
Determinant of a transformationis How much the AREA of the new Graph scaled.
JUST TO REMEMBER: THE DETERMINANT IS ABOUT AREA OF THE GRAPH!
Refer to 3Blue1Brown: The determinant
Unit vector graph
We all know the unit vector i & j made an area of 1.
But when we do a Linear transformation to the unit vector graph, the area is not 1 anymore, might be bigger or smaller.
So how much it re-sized we call it the determinant.
Note that:
- Since
a/1 = a, so calculating the Area of thenew unit vector graph, is equal to the scalar itself. - Calculating how much the
unit vector graphscaled, is exactly equal to how much the whole graph scaled.
Irregular shape
If it’s not a
grid squarecan be approximately very well by many manysmall piece of grid squares.
Determinant formula for 2x2 Matrix
It will be so much easier if you just to memorise the formula, than to understand where it comes from, which is also not necessary to do.
Determinant formula for 3x3 Matrix
I hope you’re not gonna have chance to apply this formula.
“This (determinant) is both tricky to show and derive, and is kind of pointless. Knowing how to do the operations (of determinant) isn’t a useful skill anymorebecause we just type det(A) into a computer. Thus I’ll just type det(A) and my computer gives me the answer, done. From a learning perspective, it doesn’t add much. So I’m not going to teach you how to do determinants. If you want to know, then look up a QR decomposition online, or better yet, look in a linear algebra textbook.” — David Dye, Imperial College London
Zero determinant
If the determinant of a transformation det(M) = 0, then it means the Transformation squishes the graph to a line or a point!
Negative determinant
A negative determinant means the graph has been flipped over by the transformation.
Then the j unit vector flip over to the LEFT side of i unit vector.
Refer to 3Blue1Brown for visualisation
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