Tag Archives: orthonormal vectors

Linear Algebra and Its Applications, Exercise 3.4.10

Exercise 3.4.10. Given the two orthonormal vectors and and an arbitrary vector , what linear combination of and is the least distance from ? Show that the difference between and that combination (i.e., the error vector) is orthogonal to both … Continue reading

Posted in linear algebra | Tagged , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.4.9

Exercise 3.4.9. Given the three orthonormal vectors , , and , what linear combination of and is the least distance from ? Answer: Any linear combination of and is in the plane formed by and . The combination closest to … Continue reading

Posted in linear algebra | Tagged , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.4.7

Exercise 3.4.7. Given where are orthonormal vectors, compute and show that Answer:We have so that since the transpose of a sum is equal to the sum of the transposes. The product of the sums can then be decomposed into two … Continue reading

Posted in linear algebra | Tagged | Leave a comment

Linear Algebra and Its Applications, Exercise 3.4.5

Exercise 3.4.5. Given a unit vector and , prove that is orthogonal. What is when ? Answer: We have Since the matrix is orthogonal. If then so that NOTE: This continues a series of posts containing worked out exercises from … Continue reading

Posted in linear algebra | Tagged | Leave a comment

Linear Algebra and Its Applications, Exercise 3.4.3

Exercise 3.4.3. Given the orthonormal vectors and and the vector from the previous exercise, project onto a third orthonormal vector . What is the sum of the three projections? Why? Why is the matrix equal to the identity matrix ? … Continue reading

Posted in linear algebra | Tagged , , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.4.2

Exercise 3.4.2. Given two orthonormal vectors and and the vector , project onto and . Also find the projection of onto the plane formed by and . Answer: Since is orthonormal, the projection of onto is given by Similarly the … Continue reading

Posted in linear algebra | Tagged , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.1.6

Exercise 3.1.6. What vectors are orthogonal to and in ? From these vectors create a set of three orthonormal vectors (mutually orthogonal with unit length). Answer: If is a vector orthogonal to both and then the inner product of with … Continue reading

Posted in linear algebra | Tagged , | Leave a comment