Problem 4. Let V be a vector space with ordered bases B = ( v ,..., ). Let Lp : V - R" be the coordinate isomorphisms determined by basis B. Let T : V - V be a linear transformation, and let [Tly be the n x n matrix whose j-th column is [T(v,)]B. (a) Draw a square diagram with four arrows showing the relationship between T, LB, and [Tly. Two of your arrows will represent the same map. (b) Prove that LBOT = [T]B O LB. [HINT: Two maps are equal if they have the same source/target, and they produce the same outputs for each input.] (c) Prove that there is an isomorphismim T ~ im[T]g. [HINT: Let f =( LB) imp, the restriction of Lg to im T.] (d) If C is a different ordered basisfor V, prove that im(T]s ~ im[T]c