Ryan L. answered • 01/15/14

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Experienced Engineer GTA, Making Math Concepts Easy

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|

|y'| = |0 1 ||y|

hopefully this helps