I have a confusion about transformation in linear algebra.
Consider a vector(1,1) in x,y coordinate. I have a translation 1 unit upwards. (x',y') = (1,1) + (0,1)=(1,2) this means my vector now is in (1,2) in x,y coordinate? or I am in some x',y' coordinate? If I am in a new coordinate frame x',y', my new vector should be still (1,1)? Since my new coordinate frame x',y' should be also transformed 1 unit upward.
Or this transformation is describing coordinate frame. Then my coordinate frame move 1 unit upwards. Then my new vector should be (0,-1) in x',y' , but how can I get -1 from the transformation?
Similarly in rotation, I don't quite understand the rotation matrix is rotating the vector or rotating the coordinate frame.
Thank you!
A Linear Transformation always preserves linearity in the sense that:
$$L(\sum_{i=1}^{n}\alpha_i e_i) = \sum_{i=1}^{n}L(\alpha_i e_i)$$
This is, if you input $x=(x,y)=e_1+e_2=(1,1)$, then $L(x,y)=L(1,1)=L(e_1+e_2)=L(1,0)+L(0,1)$. Then you can visualize the output as a transformation of (x,y), so this vector is $(x',y')=L(x,y)$.
This is, you got two pairs, first the origin or input point $(x,y)$ and the transformation point $(x',y')$. You can either update $(x,y)$ to be $(x',y')$ or maintain $(x,y)$ and $(x',y')$ separated. Of course, the inverse of the linear transformation (if it's a non singular matrix) would return to $(x,y)$ given the point $(x',y')$.