Before consideration of the matter it is worth reminding that any ordered system n of linearly independent vectors of space R^n is called basis of this space. At the same time the vectors forming a system will be considered as linearly independent if any their zero linear combination is possible only due to equality to zero all coefficients of this combination.
It is required to you
- - paper;
- - handle.
Instruction
1. Using only the main definitions to check linear independence of a system a vector columns, and respectively and to draw the conclusion about existence of basis, it is very difficult. Therefore in this case you can be helped by use of some special signs.
2. It is known that vectors are linearly independent if the determinant made of them is not equal to zero. Proceeding from it, it is possible to explain enough the fact that the system of vectors forms basis. So, to prove that vectors form basis, it is necessary to make determinant of their coordinates and to make sure that it is not equal to zero. Further, for reduction and simplification of records, a vector column a matrix column we will replace representation with the transposed matrix line.
3. Example 1. Whether form basis in R^3 a vector columns (1, 3, 5) ^T, (2, 6, 4) ^T, (3, 9, 0) ^T.Решение. Make determinant |A| which lines are elements of the set columns (see fig. 1). Having opened this determinant by the rule of triangles, it will turn out: |A| = 0+90+36-90-36-0=0. Therefore, these vectors cannot form basis.
4. Example. 2. The system of vectors consists from (10, 3, 6) ^T, (1, 3, 4) ^T, (3, 9, 2) ^T. May they form basis? Decision. By analogy with the first example make determinant (see fig. 2): |A| =60+54+36-54-360-6=270, i.e. is not equal to zero. Therefore, this system a vector columns is suitable for use as basis in R^3.
5. Now with all evidence it becomes clear that for finding of basis of a system the vector columns is quite enough to take any determinant of suitable dimension other than zero. Elements of its columns form a basic system. Besides, it is always desirable to have the simplest basis. As the determinant of a single matrix is always other than zero (at any dimension), as basis it is always possible to choose a system (1, 0, 0, …, 0) ^T, (0, 1, 0, …, 0) ^T, (0, 0, 1, …, 0) ^T, …, (0, 0, 0, …, 1) ^T.