This article is about linear subspaces of an abstract vector space. For subspaces of R^{n}, see Euclidean subspace. The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
Three onedimensional subspaces of R2 In linear algebra, a Euclidean subspace (or subspace of Rn) is a set of vectors that is closed under addition and scalar multiplication. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
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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. 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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
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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: â†” â‡” â‰¡ logical symbols representing iff. ...
 The zero vector, 0, is in W.
 If u and v are elements of W, then the sum u + v is an element of W;
 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. Since elements of W are necessarily elements of V, axioms 1, 2 and 58 of a vector space are satisfied a fortiori. By the closure of W under scalar multiplication (specifically by 0 and 1), axioms 3 and 4 of a vector space are satisfied. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
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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 R^{3}. Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
Proof:  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). Then u + v = (u_{1}+v_{1},u_{2}+v_{2},0+0) = (u_{1}+v_{1},u_{2}+v_{2},0). Thus, u + v is an element of W, too.
 Given u in W and a scalar c in R, if u = (u_{1},u_{2},0) again, then cu = (cu_{1}, cu_{2}, c0) = (cu_{1},cu_{2},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 R^{2}. Take W to be the set of points (x,y) of R^{2} such that x = y. Then W is a subspace of R^{2}. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ...
Proof:  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}. 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}) 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 cp = (cp_{1},cp_{2}); since p_{1} = p_{2}, then cp_{1} = cp_{2}, so cp is an element of W.
In general, any subset of an Euclidean space R^{n} 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. Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ...
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. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of R^{R}. This article is about functions in mathematics. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
Proof:  We know from calculus the sum of continuous functions is continuous.
 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. The same sort of argument as before shows that this is a subspace too. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
Examples that extend these themes are common in functional analysis. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
Properties of subspaces A way to characterise subspaces is that they are closed under linear combinations. That is, W is a subspace if and only if every linear combination of (finitely many) elements of W also belongs to W. Conditions 1 and 2 for a subspace are simply the most basic kinds of linear combinations. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
â†” â‡” â‰¡ logical symbols representing iff. ...
In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
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, all elements of B that also belong to A), but no other elements. ...
Proof:  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.
 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.
 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 In mathematics, the dimension of a vector space V is the cardinality (i. ...
For every vector space V, the set {0} and V itself are subspaces of V. 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 additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ...
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. ...
