Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually called simply a subspace when the context serves to distinguish it from other types of subspaces. In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually called simply 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 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 w 1 , w 2 {displaystyle w_{1},w_{2}} are elements of W and α , β {displaystyle alpha ,eta } are elements of K, it follows that α w 1 + β w 2 {displaystyle alpha w_{1}+eta w_{2}} is in W. Let the field K be the set R of real numbers, and let the vector space V be the real coordinate space R3.Take W to be the set of all vectors in V whose last component is 0.Then W is a subspace of V. Proof: 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: In general, any subset of the real coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace.(The equation in example I was z = 0, and the equation in example II was x = y.)Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0. Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R.Let C(R) be the subset consisting of continuous functions.Then C(R) is a subspace of RR. Proof: