Solving A x = b

A basis β for a vector space V is a linearly independent subset of V that generates
or span V. If β is a basis for V, we also say that elements of β form a basis for V.
This means that every vector from V is a finite linear combination of elements from the basis.

  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.

Theorem (Kronecker--Capelli Theorem): A system of linear algebraic equations A x = b has a solution if and only if the matrix A has the same rank as the augmented matrix \( \left[ {\bf A} \,|\,{\bf b} \right] . \)

It is obvious that the rank of the augmented matrix \( \left[ {\bf A} \,|\,{\bf b} \right] \) is greater than or equal to the rank of the matrix A. They are equal if and only if b is a linear combination of the columns of the matrix A. The last condition is equivalent to the statement that there exists a vector x such that A x = b. ■

Theorem (Fredholm matrix Theorem): A system of linear algebraic equations A x = b has a solution if and only if b is orthogonal to every solution of the homogeneous equation z A = 0, that is, their inner product is zero: \( {\bf z} \cdot {\bf b} = 0 . \) In other words, the input vector b must be orthogonal to every solution of the homogeneous adjoint equation \( {\bf A}^{\ast} {\bf y} = {\bf 0} , \) which means that \( {\bf y} \perp {\bf b} \quad \Longleftrightarrow \quad {\bf y} \cdot {\bf b} =0 . \)

Sufficiency Proof: Let r be the rank of matrix A. We can assume without any loss of generality that the first r rows of the matrix A are linearly independent. Then obviously the first r rows of the augmented matrix will be also linearly independent too. If the k-th row of the matrix A is a linear combination of the first r rows of A, then there exists a nonzero vector z such that z A = 0. According to hypothesis of theorem, \( {\bf z} \cdot {\bf b} = 0 . \) This implies that the k-th row of the augmented matrix \( \left[ {\bf A} \,|\,{\bf b} \right] \) is a linear combination of the first r rows of the augmented matrix. So ranks of A and the augmented matrix are the same. According to Kronecker-Capelli theorem, the linear system of equations A x = b is consistent.

Necessity Proof: Suppose that the linear system A x = b has a solution. Then for every \( {\bf z} \in \mathbb{C}^m \) we have the equality

\[ {\bf z} \, {\bf A} \, {\bf x} = {\bf z} \, {\bf b} . \]
If z A = 0, then z b = 0. ■
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. ■