practice-theorems

# practice-theorems - x 2 n Use the Cauchy-Schwarz nequlity...

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Here are some theorems you should try to prove. 1. Let V be a subspace of R n , ~x be a vector in R n . Show that the quantity min ~v V k ~v - ~x k is attained at the vector ~v = proj V ( ~x ). Show that this is the only vector that minimzes this quantity. 2. A set C R n is called convex if given ~x,~ y C , and a number t such that 0 t 1, the vector t~x + (1 - t ) ~ y C . (a) Show that a subspace is convex (b) Give an example of a set C that is convex but is not a subspace. (c) Show that a set C is convex if and only if given ~x,~ y C , the vector 1 2 ( ~x + ~ y ) C . 3. Let V be a subspace of R n and let P be the orthogonal projection onto the subspace. Prove the following facts: (a) P 2 = P = P T . (b) range( P ) = V and ker( P ) = V (c) I n - P is the orthogonal projection onto V . (d) If A is a matrix such that A 2 = A T = A , then there is a subspace V such that A = proj V . 4. If | ~x · ~ y | = k ~x kk ~ y k , then show that ~x and ~ y are parallel. Do not use the fact that the angle between them is a multiple of π . Although this is intuitively correct, it is not deﬁnition of parallel. 5. The length of a vector ~x in R n is deﬁned by k x k = p x 2 1 + ···
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Unformatted text preview: + x 2 n . Use the Cauchy-Schwarz nequlity to prove that the length has the following properties. (a) k x k ≥ 0. (b) k x k = 0 if and only if ~x = ~ 0. (c) k c~x k = | c |k ~x k . (d) k ~x + ~ y k ≤ k ~x k + k ~ y k . 6. Any function l from l : R n → R such that NN l ( ~x ) ≥ 0, l ( ~x ) = 0 ND If l ( ~x ) = 0, then ~x = ~ H l ( c~x ) = | c | ~x T l ( ~x + ~ y ) ≤ l ( ~x ) + l ( ~ y ) is called a norm . Show that the following functions on R n are norms. (a) l ( ~x ) = max i =1 ,...,n | x i | (b) l ( ~x ) = | x 1 | + ··· + | x n | 7. A reasonable distance function d should satisfy the following properties (a) d ( ~x,~ y ) = 0 if and only if ~x = ~ y . (b) d ( ~x,~ y ) ≥ 0 for vecx,~ y ∈ R n . (c) d ( ~x,~ y ) ≤ d ( ~x,~ z ) + d ( ~ z,~ y ), for ~x,~ y,~ z ∈ R n . If l is a norm, show that d ( ~x,~ y ) = l ( ~x-~ y ) deﬁnes a distance function. 1...
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