42. Rather than use the standard definitions of addition and scalar multiplication in R®, let these two operations be defined as shown below. a. (21, y1, z1) + (2, 92, 2) (z1 + 2, Y1 + 42, Z1+ 2) c(z, y, z) (cz, cy, 0) b. (21, Y1, z1) + (#2, Y2, z2) (0, 0, 0) c(z, y, z) (cz, cy, cz) = (%1 + ¤2 + 1, yi + y2 + 1, z1 + z2 +1) (cz, cy, cz) c. (21, Y1, z1) + (#2, Y2, 72) c(z, y, z) %D d. (21, Y1, z1) + (#2, Y2, z2) (21 + 22 + 1, y1 + y2 + 1, z1 + z2 + 1) (cz +c- 1, cy + c - 1, cz + c– 1) c(z, y, z) With each of these new definitions, is R® a vector space? Justify your answers.

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