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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
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
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
Posted in linear algebra
Tagged arithmetic mean, geometric mean, Schwarz Inequality, triangle inequality
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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
Posted in linear algebra
Tagged commutative property, distributive property, inner product
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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
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
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
Linear Algebra and Its Applications, Exercise 3.1.19
Exercise 3.1.19. State whether each of the following is true or false: (a) If the subspaces and are orthogonal, then and are also orthogonal. (b) If is orthogonal to and orthogonal to then is orthogonal to . Answer: (a) In … Continue reading