Monthly Archives: August 2011

Exercise 2.1.9. Consider the set of all nonsingular 2 by 2 matrices. Is it a vector space? How about the set of singular 2 by 2 matrices? Answer: The set of all nonsingular 2 by 2 matrices is not closed … Continue reading →

Exercise 2.1.8. Consider the following system of linear equations: Does the set of solutions form a point, line, or plane? Is it a subspace? Is it the nullspace of ? The column space of ? Answer: The system corresponds to … Continue reading →

Exercise 2.1.7. Of the following, which are subspaces of ? (a) the set of all sequences that include infinitely many zeros, e.g., (b) the set of all sequences of the form where from some point onward (c) the set of … Continue reading →

Exercise 2.1.6. Consider the equation that defines a plane in 3-space. For the parallel plane through the origin find its equation and explain whether and are subspaces of . Answer: The origin must be a solution of the equation for … Continue reading →

Exercise 2.1.5. There are eight rules for vector addition and scalar multiplication operations in a vector space: There is a zero vector (0) such that for all For each there exists one and only one vector such that for all … Continue reading →

Exercise 2.1.4. Consider the set of all 3 by 3 symmetric matrices and the set of all 3 by 3 lower triangular matrices, each of which are subspaces of the space of all 3 by 3 matrices. What is the … Continue reading →

Exercise 2.1.3. For each of the following two matrices describe the matrix’s column space and nullspace. Answer: The column space for consists of the linear combinations of its columns: The column space is therefore the x axis, i.e., the set … Continue reading →

Exercise 2.1.2. Of the following subsets of , which are subspaces and which are not? (a) the set of vectors with the first component (b) the set of vectors with the first component (c) the set of vectors and for … Continue reading →

Exercise 2.1.1. Construct the following: (a) a subset of 2-D space closed under vector addition and subtraction but not scalar multiplication (b) a subset of 2-D space closed under scalar multiplication but not vector addition Answer: (a) The set of … Continue reading →

Yesterday I posted the final worked-out exercise from chapter 1 of  Gilbert Strang’s Linear Algebra and Its Applications, Third Edition. My first post was for exercise 1.2.2 almost exact 15 months ago. The book has eight chapters and two appendices, … Continue reading →