Can someone explain me what does shear in x-direction with a scalar quantity mean? Would that mean scaling in x-direction and y-direction by some value or does it mean keeping x fixed, scaling in the y-direction?

as per the hint → Vector 𝑒1=[10] will remain the same, but why would that be? Doesn’t x shearing mean we are scaling (compressing) it by a factor of 0.5? Can someone help me to explain this

Shearing means dragging or sliding the shape in one or more directions.

In Example 2, you are asked for shearing in x direction by 0.5. So, you can imagen it as dragging the shape to the right by 0.5. But, note that the vectors with x=0 will not be affected, you can assume they are anchored to the grid.

reference: