Linearly independent vectors. Let X, y, z, …, u be vectors of a linear space .

A vector v =  x +  y +  z + … + u, where , , , …,  – real numbers, also belongs to . It is called a linear combination of vectors x, y, z, …, u.

Let a linear combination of vectors x, y, z, …, u be zero-vector, i.e.

 x +  y +  z + … + u = 0 (1)

The vectors x, y, z, …, u are called linearly independent if the equality (1) holds only for  =  =  = …=  = 0. If (1) can also hold when not all numbers , , , …,  are equal to zero then the vectors x, y, z, …, u are called linearly dependent.

Theorem. If every vector of a linear space can be represented as a linear combination of linearly independent vectors e₁, e₂, …, e_n, then d( ) = n (and consequently the vectors e₁, e₂, …, e_n form a basis in the space ).

21. Transformation of coordinates at transition to a new basis in a linear space. Theorems on transition matrix and formulas of transformation of coordinates.

Let there be two basis: e₁, e₂, …, en (old) and e’₁, e’₂, …, e’_n (new) in a n-dimensional linear since Rⁿ

e’₁ = a₁₁e₁ + a₂₁e₂ + …+a_n1e_n

e’₂ = a₁₂e₁ + a₂₂e₂ + … + a_n2e_n

…………………………………..

e_n= a_1ne₁ + a_2ne₂ + ….+a_nne_n

The matrix A= ( )

is the transition matrix from the old basis to the new basis.

Theorem: Every transition matrix A is regular, i.e. det A = 0

proof: Prove the theorem for n=2

Let {e1, e2} and {e’1, e’2} be “old” and “new” basis, and

A = transition matrix

i.e. we have e’₁ = a₁₁e₁ + a₂₁e₂ *

e’₂ = a₁₂e₁ + a₂₂e₂

Assume the contrary : detA=0, i.e. 8a₁₁a₂₂ – a₁₂a₂₁ = 0

Suppose that a₂₁≠0 or a₂₂≠0. multiply both parts of the 1^st equation of (*) on a₂₂, and multiply both parts of the 2^nd equation of (*) on (-a₂₁). Then add the obtained expressions.

a₂₂e’₁ – a₂₁e’₂ = 0 ⇒ e’₁, e’₂ are linearly dependent contradicting the hypothesis that they form a basis.

Theorem: The coordinates ξ₁, …, ξ_n and ξ’₁, …, ξ’_n are connected by

ξ₁ = a₁₁ ξ’₁ + a₁₂ ξ’₂+…+a_1n ξ’_n

ξ₂ = a₂₁ ξ’₁ + a₂₂ ξ’₂+…+a_2n ξ’_n

_{………………………………………………..}

ξ₁ = a_n1 ξ’₁ + a_n2 ξ’₂+…+a_nn ξ’_n

Which are called the formulas of transformation of coordinates.

22. Subspaces of a linear space. Linear hull of vectors. Intersection, union, sum and direct sum of subspaces.

A non-empty set formed of elements of a linear space is called a subspace of the linear space if for all and every number and .

The union of and is called the set of elements such that or . The union of and is denoted by .

The intersection of and is called the set of all elements simultaneously belonging to and . The intersection of and is denoted by .

The sum of and is called the set of all elements of kind x + y where and . The sum of and is denoted by .

The direct sum of and is called the set of all elements of kind x + y where , and . The direct sum of and is denoted by .

Theorem. Both the intersection and the sum of subspaces and are subspaces of

Theorem. The dimension of the sum of subspaces and is equal to:

If x, y, z, …, u are vectors of a linear space then all vectors , where α, β, γ, …, λ are all possible real numbers, form a subspace of the space . The set of all linear combinations of vectors is called a linear hull of the vectors x, y, z, …, u and denoted by L(x, y, z, …, u).