Problem 13b: Linear Transformations

We saw in Problem 13a, part (b), that it is possible for linear maps to map sets of points which are not lines into a single line. By our definition of linear transformation it is impossible to map a line onto a set of points which do not belong to a single line. What can we say in general about the image T(P) of a linear transformation when S = (P, L) is linear?

For the problem below, S is a linear space and S' is a near-linear space.

PROBLEM 13b: (a) Show that if V is a subspace of S, then T(V) is a linear subspace of S'. So if T is onto, then S' must be a linear space. Solution

13b: (b) Show that if T is 1-1, then T-1(V') is a linear subspace of S whenever V' is a subspace of S'. Solution

13b: (c) Show that if T is 1-1, then dim V > dim T(V). Solution

13b: (d) Show that if T is an isomorphism and V' is a subspace of S', then dim V' > dim T-1(V'). Solution

References: Batten, Batten & Beutelspacher.