What does invariant line- y axis mean? How to find the matrix of shears and stretches without memorising?

Question

I know that invariant line- y axis means that the stretch or shear is parallel to the x axis - but what does that mean??

 

Also, how can i use base vectors to find the matrix of shears and stretches without having to memorise them?

 

Thanks a lot!

Answer 1

To say that it is invariant along the y-axis means just that, as you stretch or shear by a factor of "k" along the x-axis the y-axis remains unchanged, hence invariant.

 

To explain stretches we will formulate the augmented equations as x' and y' with associated stretches Sx and Sy. The initial system resembles the following:

 

x = x

y = y

 

which forms the identity

 

|x|    | 1 0 ||x|

|y| = | 0 1 ||y|

 

augmenting this to stretch forms 

 

|x'|     |Sx 0 ||x|

|y'| =  |0 Sy ||y|

 

if one of the axes is invariant then you simply use 1 for the scale factor to have it retain it's original values.

 

Shears on the other hand augment the variant state by the invariant state and a factor "k", such that for y-invariant the augmented system forms the following:

 

x' = x + ky

y' = y

 

|x'|    |1 k ||x|
|y'| = |0 1 ||y| 

 

hopefully this helps