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# Subspaces

A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V.

In general, to show that a nonempty set W with two operations (inner operation is usually called addition and outer operation of multiplication by scalars) is a vector space one must verify the ten vector space axioms. However, if W is a subspace of a known vector space V, then certain axioms need not be verified because they are "inherited" from V. For example, commutativity property needs not to be verified because it holds for all vectors from V.

Theorem: If W is a nonempty set of vectord froma vector space V, then W is a subspace of V if and only if the following conditions hold.

• If u and v are vectors in W, then u + v is in W.
• If k is a scalar and u is a vector in W, then ku is in W. ■

Example: The span of the empty set $$\varnothing$$ consists of a unique element 0. Therefore, $$\varnothing$$ is linearly independent and it is a basis for the trivial vector space consisting of the unique element---zero. Its dimension is zero.

Example: In $$\mathbb{R}^n ,$$ the vectors $$e_1 [1,0,0,\ldots , 0] , \quad e_2 =[0,1,0,\ldots , 0], \quad \ldots , e_n =[0,0,\ldots , 0,1]$$ form a basis for n-dimensional real space, and it is called the standard basis. Its dimension is n.

Example: Let us consider the set of all real $$m \times n$$ matrices, and let $${\bf M}_{i,j}$$ denote the matrix whose only nonzero entry is a 1 in the i-th row and j-th column. Then the set $${\bf M}_{i,j} \ : \ 1 \le i \le m , \ 1 \le j \le n$$ is a basis for the set of all such real matrices. Its dimension is mn.

Example: The set of monomials $$\left\{ 1, x, x^2 , \ldots , x^n \right\}$$ form a basis in the set of all polynomials of degree up to n. It has dimension n+1. ■

Example: The infinite set of monomials $$\left\{ 1, x, x^2 , \ldots , x^n , \ldots \right\}$$ form a basis in the set of all polynomials. ■

If the vectors $$\left\{ {\bf u}_1 , {\bf u}_2 , \ldots , {\bf u}_n \right\}$$ form a basis for a vector space V, then every vector in V can be uniquely expressed in the form

${\bf v} = \alpha_1 {\bf u}_1 + \alpha_2 {\bf u}_2 + \cdots + \alpha_n {\bf u}_n$
for appropriately chosen scalars $$\alpha_1 , \alpha_2 , \ldots , \alpha_n .$$ Therefore, v determines a unqiue n-tuple of scalars $$\left[ \alpha_1 , \alpha_2 , \ldots , \alpha_n \right]$$ and, conversely, each n-tuple of scalars determines a unique vector $${\bf v} \in V$$ by using the entries of the n-tuple as the coefficients of a linear combination of $${\bf u}_1 , {\bf u}_2 , \ldots , {\bf u}_n .$$ This fact suggests that V is like the n-dimensional vector space $$\mathbb{R}^n ,$$ where n is the number of vectors in the basis for V.

Theorem: Let S be a linearly independent subset of a vector space V, and let v be an element of V that is not in S. Then $$S \cup \{ {\bf v} \}$$ is linearly dependent if and only if v belongs to the span of the set S.

Theorem: If a vector space V is generated by a finite set S, then some subset of S is a basis for V

A vector space is called finite-dimensional if it has a basis consisting of a finite
number of elements. The unique number of elements in each basis for V is called
the dimension of V and is denoted by dim(V). A vector space that is not finite-
dimensional is called infinite-dimensional.

The next example demonstrates how Mathematica can determine the basis or set of linearly independent vectors from the given set. Note that basis is not unique and even changing the order of vectors, a software can provide you another set of linearly independent vectors.

Example: Suppose we are given four linearly dependent vectors:  MatrixRank[m = {{1, 2, 0, -3, 1, 0}, {1, 2, 2, -3, 1, 2}, {1, 2, 1, -3, 1, 1}, {3, 6, 1, -9, 4, 3}}] 
Out[1]= 3

Then each of the following scripts determine a subset of linearly independent vectors:

 m[[ Flatten[ Position[#, Except[0, _?NumericQ], 1, 1]& /@ Last @ QRDecomposition @ Transpose @ m ] ]] 
Out[2]= {{1, 2, 0, -3, 1, 0}, {1, 2, 2, -3, 1, 2}, {3, 6, 1, -9, 4, 3}}

or, using subroutine

 MinimalSublist[x_List] := Module[{tm, ntm, ytm, mm = x}, {tm = RowReduce[mm] // Transpose, ntm = MapIndexed[{#1, #2, Total[#1]} &, tm, {1}], ytm = Cases[ntm, {___, ___, d_ /; d == 1}]}; Cases[ytm, {b_, {a_}, c_} :> mm[[All, a]]] // Transpose] 

we apply it to our set of vectors.

 m1 = {{1, 2, 0, -3, 1, 0}, {1, 2, 1, -3, 1, 2}, {1, 2, 0, -3, 2, 1}, {3, 6, 1, -9, 4, 3}}; MinimalSublist[m1] 
Out[3]= {{1, 0, 1}, {1, 1, 1}, {1, 0, 2}, {3, 1, 4}}

In m1 you see 1 row and n columns together,so you can transpose it to see it as column {{1, 1, 1, 3}, {0, 1, 0, 1}, {1, 1, 2, 4}}

One can use also the standard Mathematica command: IndependenceTest. ■