Basis of a system of vectors call ordered set of linearly independent vectors of e ₁, e ₂, …, en of linear system X of dimension of n. Universal the solution of a task of finding of basis of a concrete system does not exist. It is possible to calculate at first it, and then to prove existence.
It is required to you
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Instruction
1. The choice of basis of linear space can be carried out by means of the second reference given after article. You should not look for the universal answer. Pick up the system of vectors, and then give the proof of its suitability as basis. Do not try to do it algorithmically, in this case it is necessary to go some other way.
2. Any linear space, in comparison with R space³, is not rich with properties. Make addition or multiplication of a vector by number R³. It is possible to go the next way. Measure lengths of vectors and corners between them. Calculate the area size, volumes and distance between space objects. Then execute the following manipulations. Impose on any space the sklyarny work of vectors x and at ((x, y=xy +xy ₂ + … + xnyn). Now it is possible to call it Euclidean. It is of huge practical value.
3. In basis, any on dimension, enter a concept of orthogonality. If the sklyarny work of vectors x and y is equal to zero, means they ortogonalna. Such system of vectors is linearly independent.
4. Orthogonal functions generally are infinite-dimensional. Work with Euclidean functional space. Spread out on orthogonal basis of e ₁ (t), e ₂ (t), e ₃ (t), … a vector (function) x (t). Attentively study result. Find coefficient λ (coordinates of a vector x). For this purpose increase Fourier's coefficient by a vector еĸ (see the drawing). It is possible to call the formula received as a result of calculations a functional number of Fourier on the system of orthogonal functions.
5. Study the system of functions 1, sint, cost, sin2t, cos2t, …, sinnt, cosnt, …. Whether define orthogonally it on on on [-π, π]. Execute check. For this purpose calculate sklyarny works of vectors. If the result of check proves orthogonality of this trigonometrical system, then it is basis in space of C [-π, π].