What do matrices do to the vectors?

Question

Let’s consider the matrix shearing transformation. Does it change the space of the vector? So any further transformations that we do will add upon that. Or does it change the relation between vector in 2D space?

I know, it might sound as a stupid question, but I feel lost.

Answer 1

Trying to work with matrices and other coordinate-dependent constructs makes it difficult to learn the invariant (geometric) properties of vector spaces.

May I suggest starting by reading the Linear Map article on Wikipedia (it has a Russian version as well, if you prefer). You will see that a linear transformation generally acts from a vector space to a vector space, and these two spaces may be either distinct or the same one. For a deep, clean exposition, see P. Halmos's Finite-dimensional vector spaces.

For linear elastic deformations, the transformation acts from a space to itself. For an explanation of how the physical model is set up, see A. Spencer's Continuum mechanics, or at least the chapter on elasticity in Feynman's lectures on physics.

Answer 2

@avs, so my new linear map and vector space will look quite similar to this (with orange vectors as new basis ones)