Determine whether subset W x1x2 4 x1 and x1 real numbers is

Determine whether subset W = {(x_1,x_2, 4): x_1 and x_1 real numbers} is a subspace of R under the standard operations.

Solution

Solution:

According to a theorem of Vector Spaces and Subspaces, If a subset S of Rⁿ is a subspace, then the zero vector is in S.

Here the zero vector 0 = (0, 0, 0) is not in the subset W = (x_1, x₂, 4).

So W is not a subspace of R³.