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    I am having trouble with this question, could anyone give me any tips?Name:  q48.PNG
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    The conditions on L already define the whole map implicitly, since linear maps are (uniquely) determined by their action on a basis.
    So for existence, just define this map explicitly. That is, define L for any vector in R^p, noting that any such vector can be expressed as a unique linear combination of the basis vectors b_i.

    For uniqueness, just suppose that two maps L_1 and L_2 both satisfy the conditions on L given in the questions and prove that they must be equal at any vector in R^p. Again, remember that any such vector can be expressed as a linear combination of the b_i since the b_i form a basis and so span R^p.
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