QUESTION 3 3. Linear Mappings Let us consider the following linear mapping R + R° in the variables x = (r1, 12, 13, 14) f (11, 12, 73, 74) = (rı + kr2 + 14, kr1 – 13, 12 + 13 + 14) with keR parameter. a. Determine, if they exist, the values of k which make the mapping f injective, surjective, bijective For those values of k which make the mapping f not injective, determine the kernel of the b. mapping ker () (that is find a basis spanning ker (f). If multiple scenarios are possible analyse them all. For those values of k which make the mapping f not surjective, determine the image of the mapping Im (f) (that is find a basis spanning Im (f). If multiple scenarios are possible analyse them all. C. For each scenario identified (depending on the value of k), state the dimensions of kernel d. ker(f) and image Im (f) on the basis of the calculations previously carried out. Then, verify the validity of the Rank-Nullity Theorem.

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.