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Category Archives: linear algebra
Linear Algebra and Its Applications, Exercise 3.2.9
Exercise 3.2.9. Consider the projection matrix that projects onto a line. Show that . Answer: Note that is a scalar (the inner product of with itself) and is a matrix. We have NOTE: This continues a series of posts containing … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.8
Exercise 3.2.8. Consider a tetrahedon representing the methane molecule CH4, with vertices (hydrogen atoms) at , , , and , and the center (carbon atom) at . What is the cosine of the angle between the rays going from the … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.7
Exercise 3.2.7. Show that . Hint: use the Schwarz inequality with an appropriate choice of . Answer: As noted in the hint, the key to proving this is to find an appropriate choice of . The easiest way to do … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.6
Exercise 3.2.6. Suppose that and are unit vectors. Then a one-line proof of the Schwarz inequality is as follows: What previous exercise justifies the middle step of this proof? Answer: From exercise 3.2.1(a) we have for any positive and . … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.5
Exercise 3.2.5. If is a vector in then what is the angle between and the coordinate axes? What is the matrix that projects vectors in onto ? Answer: Consider the coordinate axis and the unit vector lying along that axis, … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.4
Exercise 3.2.4. Show that the Schwarz inequality is an equality if and only if and are on the same line through the origin. Describe the situation if and are on the opposite sides of the origin. Answer: We assume that … Continue reading
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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My Math Homework 