Show that the set ( S ) is a subspace of the appropriate ( ℝⁿ ) and find a basis for this subspace and its dimension. ( S ) consists of all vectors ( ⟨ 0, x, 0, 2x, 0, 3x, 0 ⟩ ) in ( ℝ⁷ ).
a) ( {Basis: }⟨ 0, 1, 0, 2, 0, 3, 0 ⟩, { Dimension: } 1 )
b) ( {Basis: }⟨ 1, 0, 0, 0, 0, 0, 0 ⟩, { Dimension: } 1 )
c) ( {Basis: }⟨ 0, 1, 0, 0, 0, 1, 0 ⟩, { Dimension: } 2 )
d) ( {Basis: }⟨ 0, 1, 0, 2, 0, 3, 0 ⟩, { Dimension: } 2 )