Thread: shear

Hey everyone... can some one explain to me why a vertical shear of 5 is NOT
1 0
0 5

thanks!

Well the general form for a vertical shear is

1 0
m 1

but you can also tell by multiplying it by the basis vectors.

[[1,0],[0,5]]*[1,0]=[1,0]

so you see the (1,0) vector does not change in the transformation, which should not be the case in a vertical shear. What this actually does is stretch everything out, because as you can see any y value is multiplied by 5, where in a shear transformation the magnitude of the y value changes according to the x value.

ok so for a vertical shear, u put the variable in the 'x' column...??
so it should be
1 0
5 1 ??

i dont quite understand why the 5 isnt in the y column

Thats correct.

think about how something in a vertical sheer moves. It moves vertically, but the magnatitude of that movement is determined by its horizontal position (the farther it's x value is from zero the more its y value changes)

so [[1,0],[5,1]]*[x,y] = [x,(5*x+y)]

so while you could say 5 is in the 'x collumn', because it is multiplied by x, it is in the 'y row', because that product is added to y*1 to become the new y value.

ok so if you have a point on the x, y plane, say (2, -1)... in matrix form it would be written as [2 -1]


then how come in my book under matrix transformation it has it written as
[2
-1]

??

In matrix form they should be put one on top of the other, but that can be hard to type on a computer so to make a matrix sometimes people will use brackets inside brackets (like how x^2 means x squared)

so [[1,2],[3,4]] would be similar to

1 2
3 4