Linear Algebra 7 | Examples for Subspaces

Subspaces - Examples with Solutions

Which of the following subsets of R2 with the usual operations of vector addition and scalar multiplication are subspaces? (T indicates Transpose)a) W1 is the set of all vectors of the form [x y], where y ≤ 0.b) W2 is the set of all vectors of the form [x y], where x ≥ 0 and y ≤ 0.c) W3 is the set of all vectors of the form [x y], where x = 0 and y = 0.

(a) Prove that the only subspaces of R^1 are R^1 and the zero subspace. (b) Prove that a subspace of R^2 is R^2, or the zero subspace, or consists of all scalar multiples of some fixed vector in R^2. (The last type of subspace is, intuitively, a straight line through the origin.) (c) Can you describe the subspaces of R^3 ? | Numerade