# Monthly Archives: March 2014

## Linear Algebra and Its Applications, Exercise 3.1.18

Exercise 3.1.18. Suppose that is the subspace of containing only the origin. What is the orthogonal complement of ()? What is if is the subspace of spanned by the vector ? Answer: Every vector is orthogonal to the zero vector. … 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 3.1.15

Exercise 3.1.15. Is there a matrix such that the vector is in the row space of the matrix and the vector is in the nullspace of the matrix? Answer: The row space of any matrix is the orthogonal complement to … 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