Linear independence criterion for rows (columns).

Transformation of coordinates when changing the basis.

Let systems of vectors e = {e 1 , …, e n } and f = {f 1 , …, f n } be two bases of the n -dimensional linear space L n .

Let x e = ( x 1 , x 2 , …, x n ) and x f = ( x’ 1 , x’ 2 , …, x’ n ) be the coordinates of the vector xL n respectively in the bases e and f .

The following is true x e = C e→f x f :

Here C e→f is the transition matrix from basis e to basis f , this is a matrix whose columns are the coordinates of the basis vectors f 1 , …, f n in the basis e 1 , …, e n :

f 1 u003d c 11 e 2 + c 21 e 1 + … + c n 1 e n , f 2 u003d c 12 e 1 + c 22 e 2 + … + c n 2 e n , …, f n = с 1 n e 2 + … + с nn e n .

The formula for transforming the coordinates of a vector when changing the basis is usually written as

x f = (C e→f ) − 1 x e

OR

Let in -dimensional linear space basis chosen , which we will call for convenience “old” and another basis , which we will call “new”. Take an arbitrary vector from . We denote its coordinate column in the old basis , and in the new . We need to find out how the coordinates in the old and in the new basis are related to each other. To do this, we first need to “connect” the old and new bases with each other. Let us write the expansions of new basis vectors in terms of the old basis

Let us compose a matrix whose columns are the coordinate columns of the vectors of the new basis

This matrix is called the transition matrix from the old basis to the new one.

Linear independence criterion for rows (columns).

Any columns included in a linearly independent system form a linearly independent subsystem.

If the column system is linearly independent, and after attaching a column to it — turns out to be linearly dependent, then the column can be sorted into columns , and, moreover, in a unique way, i.e. expansion coefficients are found uniquely.

In the previous section, the operations of matrix multiplication by a number and matrix addition were introduced, in particular, for column matrices and row matrices . Column matrices (row matrices) will be referred to below simply as columns (rows, respectively) and denoted in this chapter by capital letters. Using these operations, you can compose some algebraic expressions. Recall that columns of the same size with equal corresponding elements are considered equal.

Column is called a linear combination of columns the same size if

where – some numbers. In this case, the column is said to be sorted into columns , and the numbers are called expansion coefficients. Linear Combination with zero coefficients is called trivial .

If the columns in (3.1) are of the form

then matrix equality (3.1) corresponds elementwise equalities

The definition of a linear combination of strings of the same size is formulated similarly.

Column Set equal sizes is called a column system .

System of columns is called linearly dependent if there are such numbers , not all equal to zero at the same time, which

Here and below, the symbol o denotes the zero column of the corresponding sizes.

System of columns is called linearly independent if equality (3.2) is possible only for , i.e. when the linear combination on the left side of (3.2) is trivial. Similar definitions are formulated for rows (row matrices).

Remarks 3.1

1. One column also forms a system: is linearly dependent, and when linearly independent.

2. Any part of the column system is called subsystem .

Example 3.1. Using the definition, establish linear dependence or linear independence of column systems

Decision. 1) Columns are linearly dependent, since it is possible to compose a non-trivial linear combination, for example, with the coefficients , which is equal to the zero column: .

2) Columns are linearly independent, since the equality

equivalent to a system

turns out to be true only when .

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