Linear subspace
Encyclopedia
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 subspace
s.
(such as the field of real number
s), 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.
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 three conditions:
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 5-8 of a vector space are satisfied. 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. One can also argue that since W is nonempty, there is an element x in W, and 0 is in the field K so
and therefore property 1 is satisfied.
Example I:
Let the field K be the set R of real number
s, and let the vector space V be the Euclidean 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: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.
Take W to be the set of points (x,y) of R2 such that x = y.
Then W is a subspace of R2.
Proof: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 a Euclidean space Rn that is defined by a system of homogeneous linear equation
s 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 function
s from R to R.
Let C(R) be the subset consisting of continuous
functions.
Then C(R) is a subspace of RR.
Proof:
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.
Examples that extend these themes are common in functional analysis
.
s.
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.
Conditions 2 and 3 for a subspace are simply the most basic kinds of linear combinations.
U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V.
Proof: 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 W are vector spaces, then 0 belongs to both sets. Thus, 0 belongs to U ∩ W.
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.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and related fields of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
.
A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
s.
Definition and useful characterization and subspace
Let K be a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
(such as the field of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s), and let V be a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over K.
As usual, we call elements of V vectors and call elements of K scalars.
Suppose that W is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it satisfies the following three conditions:
- The zero vector, 0, is in W.
- If u and v are elements of W, then any linear combination of u and 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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, 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 5-8 of a vector space are satisfied. 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. One can also argue that since W is nonempty, there is an element x in W, and 0 is in the field K so
and therefore property 1 is satisfied.Example I:
Let the field K be the set R of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, and let the vector space V be the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
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:
- Given u and v in
Example II:
Let the field be R again, but now let the vector space be the Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
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:
- Let p = (
In general, any subset of a Euclidean space Rn that is defined by a system of homogeneous linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s 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.
Examples related to calculus
Example III:Again take the field to be R, but now let the vector space V be the set RR of all function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s from R to R.
Let C(R) be the subset consisting of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
functions.
Then C(R) is a subspace of RR.
Proof:
- We know from calculus that 0 ∈ C(R) ⊂ RR.
- 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.
Examples that extend these themes are common in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
.
Properties of subspaces
A way to characterize subspaces is that they are closed under linear combinationLinear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
s.
That is, a nonempty set W is a subspace if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
every linear combination of (finitely many) elements of W also belongs to W.
Conditions 2 and 3 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 intersectionIntersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V.
Proof:
- Let v and w be elements of
For every vector space V, the set {0} and V itself are subspaces of V.
If V is an inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
, then the orthogonal complement of any subspace of V is again a subspace.