22.1Question.Show that the setSgiven byS={x∈R4: 9x1−2x2+ 4x3−4x4= 0}is a subspace ofR42.2Solution.2.2.1Subspace theorem.A subset S of a vector space V over a fieldF, under the same rules for addition and multiplication byscalars, is a subspace of V if and only ifi) The vector0in V also belongs to S.ii) S is closed under vector addition, andiii) S is closed under multiplication by scalars fromF2.2.2Zero vector.As 9(0) - 2(0) + 4(0) - 4(0) = 0, the zero vector ofR4is in S. We now proceed to show the two closureaxioms.