Summary
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K. Equivalently, a nonempty subset W is a subspace of V if, whenever w1, w2 are elements of W and α, β are elements of K, it follows that αw1 + βw2 is in W. As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the zero vector space consisting of the zero vector alone and the entire vector space itself. These are called the trivial subspaces of the vector space. In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. Proof: Given u and v in W, then they can be expressed as u = (u1, u2, 0) and v = (v1, v2, 0). Then u + v = (u1+v1, u2+v2, 0+0) = (u1+v1, u2+v2, 0). Thus, u + v is an element of W, too. Given u in W and a scalar c in R, if u = (u1, u2, 0) again, then cu = (cu1, cu2, c0) = (cu1, cu2,0). Thus, cu is an element of W too. Let the field be R again, but now let the vector space V be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2. Proof: Let p = (p1, p2) and q = (q1, q2) be elements of W, that is, points in the plane such that p1 = p2 and q1 = q2. Then p + q = (p1+q1, p2+q2); since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W. Let p = (p1, p2) be an element of W, that is, a point in the plane such that p1 = p2, and let c be a scalar in R. Then cp = (cp1, cp2); since p1 = p2, then cp1 = cp2, so cp is an element of W. In general, any subset of the real coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace.
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