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Tag Archives: orthogonal vectors
Linear Algebra and Its Applications, Exercise 3.4.26
Exercise 3.4.26. In the Gram-Schmidt orthogonalization process the third component is computed as . Verify that is orthogonal to both and . Answer: Taking the dot product of and we have Since and are scalars and and are orthonormal we … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.10
Exercise 3.3.10. Given mutually orthogonal vectors , , and and the matrix with columns and , what are and ? What is the projection of onto the plane formed by and ? Answer: We have where the zero entries are the result … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.6
Exercise 3.3.6. Given the matrix and vector defined as follows find the projection of onto the column space of . Decompose the vector into the sum of two orthogonal vectors and where is in the column space. Which subspace is in? … Continue reading
Posted in linear algebra
Tagged column space, left nullspace, orthogonal subspaces, orthogonal vectors, projection matrix
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Linear Algebra and Its Applications, Exercise 3.3.3
Exercise 3.3.3. Given solve to find . Show that is orthogonal to every column in . Answer: We have or if is invertible. In this case we have so that We then have and The error vector is then The … Continue reading
Linear Algebra and Its Applications, Exercise 3.3.1
Exercise 3.3.1. a) Given the system of equations consisting of and find the least squares solution and describe the error being minimized by that solution. Confirm that the error vector is orthogonal to the column . Answer: This is a … 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 3.1.21
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 3.1.17
Exercise 3.1.17. Suppose that and are subspaces of and are orthogonal complements. Is there a matrix such that the row space of is and the nullspace of is ? If so, show how to construct using the basis vectors for … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.16
Exercise 3.1.16. Describe the set of all vectors orthogonal to the vectors and . Answer: If a vector is orthogonal to the vectors and then its inner products with those vectors must be zero, so that and This is a … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.14
Exercise 3.1.14. Given two vectors and in , show that their difference is orthogonal to their sum if and only if their lengths and are the same. Answer: First we assume that is orthogonal to . This means that their … Continue reading
Posted in linear algebra
Tagged difference of vectors, orthogonal vectors, sum of vectors
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