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Monthly Archives: May 2014
Linear Algebra and Its Applications, Exercise 3.2.12
Exercise 3.2.12. Find a projection matrix that projects every vector in onto the line described by the equation . Answer: One solution to the equation is . The projection matrix that projects vectors onto the line through is NOTE: This … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.11
Exercise 3.2.11. a) Given the line through the origin and find the matrix that projects onto this line, as well as the matrix that projects onto the line perpendicular to the original line. b) What is ? What is ? … Continue reading
Linear Algebra and Its Applications, Exercise 3.2.10
Exercise 3.2.10. Discuss whether the projection matrix from exercise 3.2.9 is invertible or not. Explain your answer. Answer:The projection matrix Note that each column of the resulting matrix is equal to the original vector multiplied by a scalar factor equal … Continue reading
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 oneline 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