Suppose TV V is a linear transformation from V to V and {v,..., Um} is a...

 

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Suppose TV V is a linear transformation from V to V and {v,..., Um} is a basis of T(V). Let 3 = {w,..., wm} CV such that T(w;) = v; and W = span{3}. Show that V = WN(T), i.e., each v V can be written uniquely as v = w + u where we W, u = N(T). Here N(T) = {x E VIT(x) = 0} is the null space. v Suppose TV V is a linear transformation from V to V and {v,..., Um} is a basis of T(V). Let 3 = {w,..., wm} CV such that T(w;) = v; and W = span{3}. Show that V = WN(T), i.e., each v V can be written uniquely as v = w + u where we W, u = N(T). Here N(T) = {x E VIT(x) = 0} is the null space. v

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To show that V W NT we need to prove two conditions V W NT For any vector v V we need to show that v ... View the full answer