Exam 3 Review PDF

Exam 3 Review Here are the chapter 4 topics on the 3rd exam. The problems are of the form “chapter.section.exercise” 1. Determine if a set V with operations +, · is a vector space (yes, be able to show all 10). 4.1.1-18... 2. Determine/show if/that a subset of a vector space is a subspace....

Exam 3 Review Here are the chapter 4 topics on the 3rd exam. The problems are of the form “chapter.section.exercise” 1. Determine if a set V with operations +, · is a vector space (yes, be able to show all 10). 4.1.1-18... 2. Determine/show if/that a subset of a vector space is a subspace. 4.2.1-5 3. Determine if a vector space is spanned by a given set of vectors. And, determine if a vector is in the span of a given set of vectors. Another way of saying this last one is “determine if a vector is a linear combination of a given set of vectors”. You should understand why those two phrases are the same. 4.2.7, 4.2.12 4. Find what a set of vectors spans. That is, write vectors as linear combinations of others. 4.2.8-10 5. Determine if vectors are (or a set of vectors is) linearly independent/dependent. 4.3.3, 4.3.9 6. Know what a basis is... Be able to determine if a set of vectors is a basis of a given vector space. Be able to take a set of vectors and add as many vectors as you need to complete it to a basis. (I believe that there was a homework question on this, and yes, guess and check is an appropriate method). 4.4.5, 4.5.12, 4.5.18, 4.5.15 7. Find the coordinates of a vector relative to a certain basis. 4.4.11, 4.4.13 8. Find the dimension of a vector space and/or subspace. 4.5.8-9 9. Understand/use the theorems involving dimension. 4.5-True/False 10. Know the “two out of 3 rule”: A set, B, of vectors forms a basis of Rn if two of the following hold: (a) B contains n vectors. (b) The span of B is Rn. (c) B is linearly independent. (you should understand why it is true as well) 11. “Transition” from one basis of a vector space to another. 4.6.3 12. Given a matrix A, find a basis for the row space, column space, and null space. 4.7.9-10 13. Use a matrix to determine a basis for the span of a set of vectors. Once you have it, what is the dimension of the span of a set of vectors? (the size of the basis). 4.5.18, 4.7.15 14. Find the rank and nullity of a matrix, A. 4.8.1, 4.8.9 15. If TA is a linear transformation, find a basis for the kernel of TA (this is the same as the null space of A). 4.10.25 16. Answer True/False questions. There are some at the end of every section for practice. Odd ones have answers in the back and ask if you have a question about an even question. That should about do it... Most importantly, relax. We’re almost done! ,