Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ =...

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉.

Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.

Expert Solution

12.

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Colby Messinger answered 2 years ago

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2 . We wish to...

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2 . We wish to prove that -> (u · v)^ 2 ≤ |u|^ 2 |v|^2 . This inequality is called the Cauchy-Schwarz inequality and is one of the most important inequalities in linear algebra. One way to do this to use the angle relation of the dot product (do it!). Another way is a bit longer, but can be considered an application of optimization. First, assume that the...

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For the following exercises, calculate u ⋅ v. Given the vectors shown in Figure 4, sketch u + v, u − v and 3v.

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show that for any two vectors u and v in an inner product space ||u+v||^2+||u-v||^2=2(||u||^2+||v||^2) give...

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Prove 1. For each u ∈ R n there is a v ∈ R n such...

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1. For this question, we define the following vectors: u = (1, 2), v = (−2,...

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2 Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and w=(3,5,4,4). 2.1 Construct a basis...

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