I am investigating for an algorithm to represent an element for example in R^n using the combination of R^(n-1),R^(n-2),...,R^1 bases(frames).In particular, is it possible to represent an element of space R^n using the base (frame) of space R^m that m
Hello Mr. Doust Azar
the answer is "NO", because every element in R^n must make by a basis that have exact n elements that every of them have the property of linear independence with the others. that's mean we cannot generate (0,0,...,0,1) with n-1 times zero with the basis of R^n-1 so we cannot generate R^n with basis of R^n-1 and so on...
Hi, dear Zarrinpour,
- I think it is possible. To illustrate, i defined a inner product and a sum in R^n that are capable to generate R^n element using the orthonormal base of R^n-1,R^n_2....R^1 .
- Best Regards
In general, one cannot represent any element of R^n by means of a base involving only elements from R^m, m < n. However, we can choose a base in R^m and complete this system of linearly independent elements from R^n up to a base in R^n. Then we can represent any element of R^n in terms of the latter base in R^n, which contains the elements of the first base of R^m.
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