The way of the proof opens directly from determination of basis. Any ordered system n of linearly independent vectors of space R^n is called basis of this space.
It is required to you
- - paper;
- - handle.
Instruction
1. Find some short sign of linear independence. Theorem. The system from t of vectors of space of R^n is linearly independent in only case when the rank of the matrix made of coordinates of these vectors is equal to t.
2. Proof. We use definition of linear independence which says that the vectors forming a system are linearly independent (in only case when) if equality to zero any their linear combination is achievable only at equality to zero all coefficients of this combination. Further see fig. 1 where everything is written most in detail. In fig. 1 in columns sets of numbers xij, j=1, 2, …, n corresponding to a vector of xi, i=1, …, are located m.
3. Perform operations by rules of linear operations in R^n space. As each vector in R^n unambiguously is defined by an ordered set of numbers, equate "coordinates" of equal vectors and receive system n of the linear uniform algebraic equations with n unknown a1, a2, …, am (see fig. 2).
4. Linear independence of a system of vectors (x1, x2, …, xm) owing to equivalent transformations is equivalent that the uniform system (fig. 2) has the only zero decision. The joint system has in only case when the only decision when the matrix rank (the matrix of a system is made of coordinates of vectors (x1, x2..., xm) systems is equal to number of unknown, that is n. So, to prove the fact that vectors form basis it is necessary to make determinant of their coordinates and to make sure that it is not equal to zero.