In each of the following, determine whether the indicated addition and scalar mu

Question

In each of the following, determine whether the indicated addition and scalar multiplication of ordered triples of real numbers yields a vector space. For those that are not vector spaces, determine which properties of a vector space fail to hold. (x1, y1, z1) + (x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2), c(x, y, z) = (cx, y, cz) (x1, y1, z1) + (x2, y2, z2) = (z1 +z2,y1 + y2, x1 + x2), c(x, y, z) = (cx, cy, cz) (x1, y1, z1) + (x2, y2, z2) = (x1 + x2, y1 + y2 - 2, z1 + z2), c(x, y, z) = (cx, y, z)

Explanation / Answer

a. fails as, C( x,y,z ) is not equal to( cx,cy,cz) b. fails , as[ x1, y2,z3] + [x2,y2,z2] is not equal to [x1 + x2 , y1 + y2 , z1+z2 ]. c. fails both the properties fail. C( x,y,z ) is not equal to( cx,cy,cz) as[ x1, y2,z3] + [x2,y2,z2] is not equal to [x1 + x2 , y1 + y2 , z1+z2 ].

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