Consider the following system: (A) Apply the elementary row operations to the corresponding augmented matrix to...

 

  
 
 

Transcribed Image Text:

Consider the following system: (A) Apply the elementary row operations to the corresponding augmented matrix to the system above in order to write it in an echelon form. (solution) (B) Determine the values of k and h such that the linear system has a specific number of solutions stated in the table given below and justify briefly your answers. (solution) # of solutions (i) no solution (ii) unique solution (iii) many solutions (kx + y + z = 0 X + z = 1 (x+y=z=h k h brief justification (C) When the system has many solutions in (B)-(iii) above, find the general solution of the system in parametric vector form. (D) Give a full geometric description of the solution set in (C) above. (solution) [k (E)-(P) Let A be the coefficient matrix of the system with many solutions. That is, A = 1 [1 you found in (B)-(iii) above. (E) Do the columns in A span R²? Explain why or why not. (solution) 1 0 1 1 1, where k is the value -1 (F) Do the columns in A span R³? Explain why or why not. (solution) (G) Give a full geometric description of the set spanned by all the columns in A. (solution) (H) Are the columns in A linearly independent? Explain why or why not. (solution) (1) If your answer is 'Yes', then skip this question. Otherwise, find the dependence relation of the columns. (solution) (J) Suppose that d = 2. Find the solution set of Au = d. Use Linear Algebra, not Elementary Algebra. [3] (solution) = b consistent for all b in R³? Explain why or why not. (K) Is Au (solution) (L)-(M) If your answer in (K) above is 'Yes', then skip the following two questions, (L) and (M). Otherwise, answer them. (L) Find the parametric vector form of elements in the set of all b that makes the system Au (solution) = b consistent. (M) Give a full geometric description of the set in (L) above. (solution) (N)-(P) Let T: R³ → R³ be a matrix transformation defined by T(v) = Av. (N) Show that the transformation is linear by the definition. (solution) (0) Determine if the transformation is onto. Justify your answer briefly. (solution) (P) Determine if the transformation is one-to-one. Justify your answer briefly. (solution) Consider the following system: (A) Apply the elementary row operations to the corresponding augmented matrix to the system above in order to write it in an echelon form. (solution) (B) Determine the values of k and h such that the linear system has a specific number of solutions stated in the table given below and justify briefly your answers. (solution) # of solutions (i) no solution (ii) unique solution (iii) many solutions (kx + y + z = 0 X + z = 1 (x+y=z=h k h brief justification (C) When the system has many solutions in (B)-(iii) above, find the general solution of the system in parametric vector form. (D) Give a full geometric description of the solution set in (C) above. (solution) [k (E)-(P) Let A be the coefficient matrix of the system with many solutions. That is, A = 1 [1 you found in (B)-(iii) above. (E) Do the columns in A span R²? Explain why or why not. (solution) 1 0 1 1 1, where k is the value -1 (F) Do the columns in A span R³? Explain why or why not. (solution) (G) Give a full geometric description of the set spanned by all the columns in A. (solution) (H) Are the columns in A linearly independent? Explain why or why not. (solution) (1) If your answer is 'Yes', then skip this question. Otherwise, find the dependence relation of the columns. (solution) (J) Suppose that d = 2. Find the solution set of Au = d. Use Linear Algebra, not Elementary Algebra. [3] (solution) = b consistent for all b in R³? Explain why or why not. (K) Is Au (solution) (L)-(M) If your answer in (K) above is 'Yes', then skip the following two questions, (L) and (M). Otherwise, answer them. (L) Find the parametric vector form of elements in the set of all b that makes the system Au (solution) = b consistent. (M) Give a full geometric description of the set in (L) above. (solution) (N)-(P) Let T: R³ → R³ be a matrix transformation defined by T(v) = Av. (N) Show that the transformation is linear by the definition. (solution) (0) Determine if the transformation is onto. Justify your answer briefly. (solution) (P) Determine if the transformation is one-to-one. Justify your answer briefly. (solution)

Answer rating: 100% (QA)

A To write the system in echelon form well perform elementary row operations on the augmented matrix ... View the full answer