The fact that for any linear map the rank plus the nullity equals the dimension of the domain shows that a necessary condition for the existence of a homomorphism between two spaces, onto the second space, is that there be no gain in dimension.
That is, where h: V †’ W is onto, the dimension of W must be less than or equal to the dimension of V.
(a) Show that this (strong) converse holds: no gain in dimension implies that there is a homomorphism and, further, any matrix with the correct size and correct rank represents such a map.
(b) Are there bases for R3 such that this matrix

represents a map from R3 to R3 whose range is the xy plane subspace of R3?

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