The goal of this course was to solve linear systems. Therefore, each chapter should provide something to help with this goal. Below is a short answer section which shows how each chapter can be used. 2x + y + z = 1 5x + 3y + z = 2. r + y + z = 0 (*) Consider the system (a) Use gaussian elimiation to solve the system above. Hint : Recall that we were able to solve these without the method from part (b). Check the first few days of notes. (b) Give the matrix equation A = i for the system above. Now use gauss - jordan elimination on (A | 5) to solve. (c) Compute the determinant of the matrix A from (b). Use this to show that A = b has a solution. (d) Use the column space theorem to conclude A = b, has a solution. (e) Using A, construct a linear map T : R³ → R³. (f) If T : R* → R® (from the above) is onto, then given be codom(T), there exists T € dom(T) such that T(F) = b. Show that T is onto. Use this to conclude that A = b has a solution. (g) Suppose A = PDP-' and we wanted to solve A = b. Given A, this equation becomes, A = b = (PDP-')# = b. If we set j = P-'ë then we can first solve PDj = b. Using the j from this equation, we're done. Why? As j = P¯'ï, we have = Pj. Does this make sense?

The goal of this course was to solve linear systems. Therefore, each chapter should provide something to help with this goal. Below is a short answer section which shows how each chapter can be used. 2x + y + z = 1 5x + 3y + z = 2. r + y + z = 0 () Consider the system (a) Use gaussian elimiation to solve the system above. Hint : Recall that we were able to solve these without the method from part (b). Check the first few days of notes. (b) Give the matrix equation A = i for the system above. Now use gauss - jordan elimination on (A | 5) to solve. (c) Compute the determinant of the matrix A from (b). Use this to show that A = b has a solution. (d) Use the column space theorem to conclude A = b, has a solution. (e) Using A, construct a linear map T : R³ → R³. (f) If T : R → R® (from the above) is onto, then given be codom(T), there exists T € dom(T) such that T(F) = b. Show that T is onto. Use this to conclude that A = b has a solution. (g) Suppose A = PDP-' and we wanted to solve A = b. Given A, this equation becomes, A = b = (PDP-')# = b. If we set j = P-'ë then we can first solve PDj = b. Using the j from this equation, we're done. Why? As j = P¯'ï, we have = Pj. Does this make sense?

Name: I need these ones solved: (e, f and g) The goal of this course was to
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Description: I need these ones solved: (e, f and g) The goal of this course was to solve linear systems. Therefore, each chapter shouldprovide something to help with this goal. Below is a short answer section whichshows how each chapter can be used.2x + y + z = 1