Equivalently, subspaces can be characterized by the property of being closed under linear combinations. The same sort of argument as before shows that this is a subspace too.Įxamples that extend these themes are common in functional analysis.įrom the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Keep the same field and vector space as before, but now consider the set Diff( R) of all differentiable functions.
We know from calculus that 0 ∈ C( R) ⊂ R R.Let C( R) be the subset consisting of continuous functions. Geometrically, these subspaces are points, lines, planes and spaces that pass through the point 0.Īgain take the field to be R, but now let the vector space V be the set R R of all functions from R to R. (The equation in example I was z = 0, and the equation in example II was x = y.) In general, any subset of the real coordinate space R n that is defined by a system of homogeneous linear equations will yield a subspace. Then c p = ( cp 1, cp 2) since p 1 = p 2, then cp 1 = cp 2, so c p is an element of W. Let p = ( p 1, p 2) be an element of W, that is, a point in the plane such that p 1 = p 2, and let c be a scalar in R.Then p + q = ( p 1+ q 1, p 2+ q 2) since p 1 = p 2 and q 1 = q 2, then p 1 + q 1 = p 2 + q 2, so p + q is an element of W. Let p = ( p 1, p 2) and q = ( q 1, q 2) be elements of W, that is, points in the plane such that p 1 = p 2 and q 1 = q 2.Take W to be the set of points ( x, y) of R 2 such that x = y. Let the field be R again, but now let the vector space V be the Cartesian plane R 2.
Given u in W and a scalar c in R, if u = ( u 1, u 2, 0) again, then c u = ( cu 1, cu 2, c0) = ( cu 1, cu 2,0).Given u and v in W, then they can be expressed as u = ( u 1, u 2, 0) and v = ( v 1, v 2, 0).Take W to be the set of all vectors in V whose last component is 0. Let the field K be the set R of real numbers, and let the vector space V be the real coordinate space R 3. These are called the trivial subspaces of the vector space. Īs 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. Equivalently, a nonempty subset W is a subspace of V if, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W. 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. 6.6 Basis for the sum and intersection of two subspaces.5 Operations and relations on subspaces.