I am able to prove the above-asked using the fact that a linear map preserves linear dependence. I also vaguely suspect a connection from group theory (homomorphisms?). But I'm having trouble getting an intuition for that fact that a linear transformation cannot increase the dimension of a vector space. Help?
Linear Algebra – Why Linear Transformation Can Only Preserve or Reduce Dimension
linear algebramatrices
Related Question
- [Math] Linear transformations that preserve orthogonality
- [Math] Showing that a linear transformation is injective, surjective or bijective.
- Does a transformation that doesn’t preserve origin, lines, parallelism automatically isn’t a linear transformation
- How do linear transformation properties preserve vector space structure
Best Answer
My intuition would be this: linear maps preserve lines (that's the intuition behind them, anyway). If they are degenerate, the lines might collapse to points, but points which are colinear remain such after the transformation.
Dimension of a space is, intuitively, the number of independent lines you can draw in a space. Because linear transformation preserves not just lines, but also linear subspaces of higher dimensions (so coplanar points remain coplanar etc.), it can't "split" a line into more of them, even if it can join some of them, and it can't turn lines which weren't independent into ones which are (because that would be "splitting" some higher-dimensional subspace; it's a little circular, but we're talking about the intuition here), so the dimension can't go up.