Chapter 4 1. Identify (a) equation(s) and (b) a basis for a specific one-dimensional subspace of R3. 2. Identify (a) equation(s) and (b) a basis for a specific two-dimensional subspace of R3. 3. Identify (as above) two subspaces of RS which are nested (one fits inside the other). The nesting should be obvious in both the respective equation(s) and the respective bases. 4. (a) Identify (as above) two subspaces of R's which intersect only at one point. (b) Is it possible for two subspaces of R3 not to intersect at all? 5. Identify (as above) two subspaces of R3 which intersect at more than just a single point. The intersection should be obvious in the respective bases. 6. Identify (as above) three subspaces of RS which intersect at more than just a single point. The intersection should be obvious in the respective bases. 7. What is the nullspace of an invertible matrix? 8. Give a matrix with a nullspace of dimension (a) 0, (b) 1, (c) 2, (d) 3. 9. Identify (a) equation(s), (b) a generic form/representative and (c) a basis for a specific one- dimensional subspace of P2. (Start by imposing a "zero condition" for some combination of values of f and its derivatives.) 10. Identify (a) equation(s), (b) a generic form/representative and (c) a basis for a specific two- dimensional subspace of P2. 11. Identify (as above) two subspaces of P2 which are nested (one fits inside the other). The nesting should be obvious in both the respective equation(s) and the respective bases. 12. Identify (a) a generic form/representative and (b) a basis for a specific one-dimensional sub- space of M2x2. 13. Identify (a) a generic form/representative and (b) a basis for a specific two-dimensional sub- space of M2x2. 14. Identify (as above) two subspaces of M2x2 which are nested (one fits inside the other). The nesting should be obvious in both the respective form(s) and the respective bases. 15. True or false: Any 2 vectors in R2 that are not parallel span the whole r-y plane. 16. True or false: Considering a matrix B as its columns augmented together, B = [61 162| . . . (5n], then its product with any matrix A can be viewed similarly: AB = [Abj | Ab2| . . . [Ab,]. 17. True or false: Any row of a matrix A is perpendicular to every vector in the nullspace of A. 18. True or false: Every vector space has infinitely many subspaces