Question: Call a matrix representation optimally sparse if it consists of zeros and ones only and the number of ones is as small as possible. Let

Call a matrix representation optimally sparse if it consists of zeros and ones only and the number of ones is as small as possible. Let V=P4(R), and let TL(V,V):ff(x+2)-2f(x+1)+f(x).

Scenario 1: Given the freedom to use any two bases for V as a domain and a co-domain, does T have an optimally sparse representation? If so, find it.

Scenario 2: Given the freedom to use any single basis for V both as a domain an a co-domain, does T have an optimally sparse representation? If so, find it.

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