Final answer:
The first function given is a linear transformation as it satisfies both properties required for a linear transformation. However, the second function is not a linear transformation because it fails to satisfy these properties when summing up elements in different rows and when multiplying by a scalar.
Step-by-step explanation:
Assessing Potential Linear Transformations
To determine if the given functions are linear transformations, we must verify that they satisfy two properties: (1) T(v + w) = T(v) + T(w) for any vectors v and w, and (2) T(cv) = cT(v) for any vector v and scalar c.
(a) Consider the transformation T:R^3→R^2 defined by T(abc) → (a+b+3c, 0). To test the first property, let u = (a1, b1, c1) and v = (a2, b2, c2) in R^3, and we calculate T(u+v). This equals T(a1+a2, b1+b2, c1+c2) = (a1+a2+b1+b2+3(c1+c2), 0) which is (a1+b1+3c1, 0) + (a2+b2+3c2, 0) = T(u) + T(v), satisfying property (1). For the second property, let c be any scalar, then T(cu) = T(ca1, cb1, cc1) = (ca1+cb1+3cc1, 0) = c(a1+b1+3c1, 0) = cT(u), satisfying property (2). Therefore, T is a linear transformation.
(b) For the transformation T: Mat_2x2(R) → R^2 defined by T((a11,a21,a12,a22)) → (a11+a12,a21+a22), consider two matrices A and B and a scalar k. Applying the definition, T(A + B) does not necessarily equal T(A) + T(B), as the transformation sums up elements in different rows, violating property (1). Similarly, T(kA) does not necessarily result in kT(A), violating property (2). Hence, T is not a linear transformation.