Let n be an even integer.In both parts of this problem,let Vbe the subspace of all vector in such that .Consider the basis of V with
a.Show that is orthogonal to
b.Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix.
a.The solution is orthogonal.
b.It suffices to show that M and N are Hankel matrices.Indeed and for all
Two vectors and in are called orthogonal if
Assume n be an even integer.
let V be the subspace of all vector in such that .Consider the basis of V with:
Note that ab=-1
Now we have to prove that for the verification of orthogonality.
Further calculation can be as follows:
Thus for all even integer n.
Hence and are orthogonal.
Therefore, the solution.
By a theorem we know that a subspace V of with orthonormal basis .The matrix P of the orthogonal projection onto V is P=QQT.
Note that the matrix P is symmetric, since:
Thus, the preceding paragraph P is a linear combination of the matrices and .
It suffices to show that M and N are Hankel matrices.
Indeed and for all
Hence the explanation.
Leonardo da Vinci and the resolution of forces. Leonardo (1452–1519) asked himself how the weight of a body, supported by two strings of different length, is apportioned between the two strings.
Three forces are acting at the point D: the tensions and in the strings and the weight . Leonardo believed that
Was he right? (Source: Les Manuscripts de Léonard de Vinci, published by Ravaisson-Mollien, Paris, 1890.)
Hint: Resolve into a horizontal and a vertical component; do the same for . Since the system is at rest, the equation holds. Express the ratios
and . In terms of and , using trigonometric functions, and compare the results.
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