Final answer:
The given functions Y1 and Y2 are not linearly dependent because Y2 includes a mixed product term that does not appear in Y1 and cannot be obtained by multiplying any term in Y1 by a constant.
Step-by-step explanation:
To determine whether the functions Y1 = 2X¹ + 3X² and Y2 = 4X¹ + 12X¹X² + 9X² are linearly dependent, we analyze if one function can be expressed as a scalar multiple of the other. In this case, we see that Y2 has a term with a product of the variables X¹X² which does not appear in Y1. Hence, Y2 cannot be represented simply as a scalar multiple of Y1 and, therefore, they are not linearly dependent.
A set of functions is linearly dependent if there exist constants, not all zero, such that a linear combination of these functions equals to zero. In this case, since Y2 contains the mixed term 12X¹X², which cannot be produced by any scalar multiple of any of the terms in Y1, there are no constants "c1" or "c2" such that "c1Y1 + c2Y2 = 0" for all values of X¹ and X². Therefore, the functions are linearly independent.