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Exercise 15

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Because the rank of a matrix equals the rank of any map it represents, if one matrix represents two different maps H=RepB,D(h)=RepB^,D^(h^)H={\rm Rep}_{B,D}(h)={\rm Rep}_{\hat{B},\hat{D}}(\hat{h}) (where h,h^ ⁣:V→Wh,\hat{h}\colon V\to W) then the dimension of the range space of hh equals the dimension of the range space of h^\hat{h}. Must these equal-dimensional range spaces actually be the same?

Hefferon, Linear Algebra, Maps Between Spaces, Any Matrix Represents a Linear Map, Exercise 15

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.