Kunze Linear Algebra · Problem 3.2.10

Linear Transformations

Mathematics — Methods · onto-nonsingular-rectangular

Problem

Let AA be an m×nm \times n matrix with entries in FF and let TT be the linear transformation from Fn×1F^{n \times 1} into Fm×1F^{m \times 1} defined by T(X)=AXT(X) = AX. Show that if m<nm < n it may happen that TT is onto without being non-singular. Similarly, show that if m>nm > n we may have TT non-singular but not onto.

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