hw3 - 5 Show that if f R n → R is a function that...

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Home Work 3 Due on October 18, 2010 Reading assignments don’t have to be turned in. 1. Reading assignment. Read chapters 2 and 3 of the class notes posted on the class web-site. 2. Show that X is a full-column rank matrix if and only if Xz = 0 has only the trrivial solution z = 0. 3. Show (in complete detail) that X is a full column rank matrix if and only if X T X is non-singular (invertible). Assume X is a real matrix. 4. Show how to construct at least one left-inverse for a full column rank matrix, and one right-inverse for a full row rank matrix.
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Unformatted text preview: 5. Show that if f : R n → R is a function that satisfies the following conditions − f ( v ) ≥ 0 for all v ∈ R n − f ( v ) = 0 iff v = 0 − f ( αv ) = | α | f ( v ) for all α ∈ R and all v ∈ R n − The set { v : f ( v ) ≤ 1 } is convex then f defines a norm on R n . 6. Show that for p ≥ 1 and p − 1 + q − 1 = 1, k x k p = max y 6 =0 | y T x | k y k q , x ∈ R n . Hint : You can use Hölder’s inequality for part of the proof....
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