Show that the following are equivalent for a linear transfor

Show that the following are equivalent for a linear transformation T: V rightarrow W. ker T = V im T = {0} T = 0 Let A and B be m Times n and k Times n matrices, respectively. Assume that Ax = 0 implies Bx = 0 for every n-column x. Show that rank A greaterthanorequalto rank B.

Solution

11) Consider the statement a.

Ker T = V

i.e. for all x in V, T(v) = identity element of W

Hence Image of T = identity element of W = {0}

Thus a and b are equivalent.

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c) Since image of T =0

matrix for transformation T is the zero matrix

So T=0

Thus a,b and c are equivalent.