The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. Linear algebra is the branch of Mathematics concerned with Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition

Definition and useful characterization

Let K be a field (such as the field of real numbers), and let V be a vector space over K. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added As usual, we call elements of V vectors and call elements of K scalars. Suppose that W is a subset of V. If W is a vector space itself, with the same vector space operations as V has, then it is a subspace of V.

To use this definition, we don't have to prove that all the properties of a vector space hold for W. Instead, we can prove a theorem that gives us an easier way to show that a subset of a vector space is a subspace.

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following 3 conditions:

1. The zero vector, 0, is in W. ↔
2. If u and v are elements of W, then the sum u + v is an element of W;
3. If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

Proof: Firstly, property 1 ensures W is nonempty. Looking at the definition of a vector space, we see that properties 2 and 3 above assure closure of W under addition and scalar multiplication, so the vector space operations are well defined. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Since elements of W are necessarily elements of V, axioms 1, 2 and 5-8 of a vector space are satisfied a fortiori. A B By the closure of W under scalar multiplication (specifically by 0 and -1), axioms 3 and 4 of a vector space are satisfied.

Conversely, if W is subspace of V, then W is itself a vector space under the operations induced by V, so properties 2 and 3 are satisfied. By property 3, -w is in W whenever w is, and it follows that W is closed under subtraction as well. Since W is nonempty, there is an element x in W, and is in W, so property 1 is satisfied.

Examples

Examples related to analytic geometry

Example I: Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R³. In Mathematics, the real numbers may be described informally in several different ways Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.

Proof:

1. 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.
2. 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.

Example II: Let the field be R again, but now let the vector space be the Euclidean geometry R². Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Take W to be the set of points (x,y) of R² such that x = y. Then W is a subspace of R².

Proof:

1. 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.
2. 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 an Euclidean space Rⁿ that is defined by a system of homogeneous linear equations will yield a subspace. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable (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.

Examples related to calculus

Example III: Again 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 Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Let C(R) be the subset consisting of continuous functions. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Then C(R) is a subspace of R^R.

Proof:

1. We know from calculus the sum of continuous functions is continuous.
2. Again, we know from calculus that the product of a continuous function and a number is continuous.

Example IV: Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The same sort of argument as before shows that this is a subspace too.

Examples that extend these themes are common in functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article

Properties of subspaces

A way to characterise subspaces is that they are closed under linear combinations. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics That is, W is a subspace if and only if every linear combination of (finitely many) elements of W also belongs to W. ↔ In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Conditions 1 and 2 for a subspace are simply the most basic kinds of linear combinations.

Operations on subspaces

Given subspaces U and W of a vector space V, then their intersection U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently

Proof:

1. Let v and w be elements of U ∩ W. Then v and w belong to both U and W. Because U is a subspace, then v + w belongs to U. Similarly, since W is a subspace, then v + w belongs to W. Thus, v + w belongs to U ∩ W.
2. Let v belong to U ∩ W, and let c be a scalar. Then v belongs to both U and W. Since U and W are subspaces, cv belongs to both U and W.
3. Since U and V are vector spaces, then 0 belongs to both sets. Thus, 0 belongs to U ∩ W.

Furthermore, the sum

is also a subspace of V. The dimensions of U ∩ W and U + W satisfy

For every vector space V, the set {0} and V itself are subspaces of V. In Mathematics, the dimension of a Vector space V is the cardinality (i

If V is an inner product space, then the orthogonal complement of any subspace of V is again a subspace. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In the mathematical fields of Linear algebra and Functional analysis, the orthogonal complement W^\bot of a subspace W

External links

• MIT Linear Algebra Lecture on the Four Fundamental Subspaces at Google Video, from MIT OpenCourseWare
• Vector subspace on PlanetMath. PlanetMath is a free, collaborative online Mathematics Encyclopedia.