Matrix Spaces

There are two kinds of matrix spaces: one is defined by a set of matrices of special type (considered in section), and another is associated with every matrix. This section is devoted to the latter case.

We know from previous discussions that every real or complex n-dimensional vector space is isomorphic to ℝn or ℂn. This gives us the ground to consider linear transformations from one Euclidean Vector Space ℝn (or ℂn) into another one ℝm (or ℂm) instead of general finite dimensional vector spaces. Every linear transformation T: V ↦ W from one n-dimensional vectors space V into another m-dimensional vector space W is equivalent to a corresponding matrix transformation x ↦ Ax for some m×n matrix A for appropriate chosen bases. The inverse statement is obviously true because multiplication by a matrix defines a linear transformation.

Let A be an m-by-n matrix that maps \( \mathbb{R}^n \) into \( \mathbb{R}^m . \) So for every (colomn) vector \( {\bf x} = \left[ x_1 , x_2 , \ldots , x_n \right]^{\mathrm{T}} \) corresponds a vector \( {\bf A}\,{\bf x} \in \mathbb{R}^m . \)

Let V and U be vector spaces and \( T:\,U \to V \) be linear transformation. Then the set of all vectors of the form \( {\bf y} = {\bf A}\,{\bf x} \in V \) is called the range (or image) of T.

Theorem: Let V and U be vector spaces and \( T:\,U \to V \) be linear transformation. Then the range of T is a subspace of V.

To clarify the notation, we use the symbols 0U and 0V to denote the zero vectors of U and V, respectively. We also denote by R(T) the range of linear transformation T.

Because \( T\left( 0U \right) = 0V, \) we have that \( 0V \in R(T) . \) Now let us choose two vectors \( {\bf x}, \, {\bf y} \in R(T) \) from the range and a scalar k. Then there exist two vectors u and v such that \( T({\bf u}) = {\bf x} \) and \( T({\bf v}) = {\bf y} . \) Hence, \( T({\bf u} + {\bf v}) = T({\bf u}) + T({\bf v}) = {\bf x} + {\bf y} \) and \( T(k{\bf u}) = k\, T({\bf u}) = k\, {\bf x} . \) Thus, \( T({\bf u} + {\bf v}) = T({\bf u}) + T({\bf v}) = {\bf x} + {\bf y} \in R(T) \) and \( T(k\,{\bf u}) = k \, T({\bf x}) = k\,{\bf x} \in R(T) , \) so R(T) is a subspace of V.

  Recall that a set of vectors β is said to generate or span a vector space V if every element from V can be represented as a linear combination of vectors from β.

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. ■

 

Theorem: Let V be a vector space and \( \beta = \left\{ {\bf u}_1 , {\bf u}_2 , \ldots , {\bf u}_n \right\} \) be a subset of V. Then β is a basis for V if and only if each vector v in V can be uniquely decomposed into a linear combination of vectors in β, that is, 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 unique scalars \( \alpha_1 , \alpha_2 , \ldots , \alpha_n . \) ■

  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. ■