![]() ![]() ![]() 1-subspaces specified by two vectors are equal if and only if one vector can be obtained from another with scalar multiplication: Geometrically (especially over the field of real numbers and its subfields), a subspace is a flat in an n-space that passes through the origin.Ī natural description of a 1-subspace is the scalar multiplication of one non- zero vector v to all possible scalar values. The same is true for subspaces of finite codimension (i.e., subspaces determined by a finite number of continuous linear functionals).ĭescriptions of subspaces include the solution set to a homogeneous system of linear equations, the subset of Euclidean space described by a system of homogeneous linear parametric equations, the span of a collection of vectors, and the null space, column space, and row space of a matrix. In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. 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. Again, we know from calculus that the product of a continuous function and a number is continuous.We know from calculus that the sum of continuous functions is continuous. ![]() We know from calculus that 0 ∈ C( R) ⊂ R R.Let C( R) be the subset consisting of continuous functions. (The equation in example I was z = 0, and the equation in example II was x = y.)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. 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. A linear subspace is usually simply called 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 5 × 5 square) is pictured four times for a better visualization The origin (0, 0), marked with green circles, belongs to any of six 1-subspaces, while each of 24 remaining points belongs to exactly one a property which holds for 1-subspaces over any field and in all dimensions. One-dimensional subspaces in the two-dimensional vector space over the finite field F 5. ![]()
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