Monthly Archives: April 2014

Linear Algebra and Its Applications, Exercise 3.2.3

Exercise 3.2.3. Find the multiple of the vector that is closest to the point . Also find the point on the line through that is closest to . Answer: The first problem amounts to finding the projection of onto . … Continue reading

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Linear Algebra and Its Applications, Exercise 3.2.2

Exercise 3.2.2. Use the formula (where is the projection of on ) to confirm that (where is the angle between and ). Answer: Since we can compute the square of the length of as Since is a scalar quantity we … Continue reading

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Linear Algebra and Its Applications, Exercise 3.2.1

Exercise 3.2.1. a) Consider the vectors and where and are arbitrary positive real numbers. Use the Schwarz inequality involving and to derive a relationship between the arithmetic mean and the geometric mean . b) Consider a vector from the origin … Continue reading

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Commutative and distributive properties for vector inner products

As I think I’ve previously mentioned, one of the minor problems with Gilbert Strang’s book Linear Algebra and Its Applications, Third Edition, is that frequently Strang will gloss over things that in a more rigorous treatment really should be explicitly … Continue reading

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Linear Algebra and Its Applications, Exercise 3.1.22

Exercise 3.1.22. Consider the equation and the subspace of containing all vectors that satisfy it. Find a basis for , the orthogonal complement of . Answer:  is the nullspace of the linear system where Since is the nullspace of its … Continue reading

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Linear Algebra and Its Applications, Exercise 3.1.21

Exercise 3.1.21. If is the plane in described by what is the equation for the plane parallel to through the origin? What is a vector perpendicular to ? Find a matrix for which is the nullspace, and a matrix for … Continue reading

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Linear Algebra and Its Applications, Exercise 3.1.20

Exercise 3.1.20. Suppose is a subspace of . Show that . What does this mean? Answer: We first consider the case where ; in other words, contains only the zero vector. From exercise 3.1.18 we know that . The only … Continue reading

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