Feedback 2 2B Linear Algebra 2017 . Feedback Exercise 2 These questions relate to the material in Parts 3-4 and Exercise Sheets 3-4. This feedback exercise is due by 3pm on Wednesday 18th October 2017. Late submissions will not be accepted. 1 The form for submitting written work, along with the continuation sheets, can be found on the 2B moodle page. Please do not staple the pages together but use paper clips instead. This work should be submitted to the appropriate box in the foyer of the Mathematics building by the deadline above. One of the aims of this course is to develop mathematical writing skills. These will be assessed explicitly in the feedback exercises.2 a) Consider the matrix 3 A = 1 1 2 1 0 1 1 . 1 Use the Gauss-Jordan method to find A1 . b) Consider the matrix 1 2 B= 1 1 2 2 1 0 1 2 . 5 3 i) Find a basis for the row space of the matrix B. ii) Use your answer to (i) and the Rank Theorem to show that nullity( B) = 1. iii) Find a basis for the null space of the matrix B. c) Let C be an m n matrix with real entries. For vectors u , v Rn , we define u v if the vector u v is in null(C ). Prove that for all u , v , w Rn : i) u u ; ii) if u v then v u ; iii) if u v and v w then u w ; and u = Cvv . iv) u v if and only if Cu That is, you are being asked to prove the following statements, and you will be marked correct for proving the following statements for all u , v , w Rn : i) the vector u u is in null(C ); We encourage you to talk to each other about mathematics, and work in groups to understand the material in this course and the exercises. However, your final submitted feedback exercise must be your own work. This means that it is suitable to talk about the problems and get the ideas together, but you should do the final write up alone. You also should not show another student your final submission or look at someone elses. 2 Mathematical writing will be assessed through a negative marking scheme with marks deducted from your overall score for making various 'standard mistakes', such as not writing in sentences. A full explanation of how the mark scheme works can be found on moodle. 1 2b linear algebra ii) if u v is in null(C ) then v u is in null(C ); iii) if u v is in null(C ) and v w is in null(C ) then u w is in null(C ); and u = Cvv . iv) u v is in null(C ) if and only if Cu 2