Homework for §4.2
Show that the set W given below is a subspace of R3 , or show that it is not by providing an
example where it fails to be closed under vector addition or scalar multiplication.
1. W = {(x, y, z) : x + 3y + 2z = 0 and − 2y + z = 3x}
2. W = {(x, y, z) : x + 2y + z = 4 and 3x + 2y = 0}
3. W = {(x, y, z) : xyz = 0}
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4. Write the solution space to the homogeneous system of equations
2x1 + 4x2 − 2x3 + 4x4 = 0
3x1 + 5x2 − 2x3 − x4 = 0
as all linear combinations of the form s~u + t~v for an appropriate choice of ~u and ~v.