Problem 2. Consider the following system of linear equations: x_(1)+3x_(3)-2x_(4)+x_(5),=,0 -2x_(1)+x_(2)+x_(4)-3x_(5),=,0 3x_(3)+2x_(4)-x_(5),=,15 6x_(3)+3x_(4)-5x_(5),=,5 2x_(1)+6x_(3)-8x_(4)+10x_(5),=,0 Part A: (i) Write this system of linear equations in matrix form Ax=b. (ii) Write down the augmented matrix A|b. (iii) Solve the system of linear equations given above by using row reduction until A|b is in reduced row-echelon form. You will need scrape paper for this. (iv) Is the coefficient matrix A nonsingular? Part B: Consider the column vectors v_(1),v_(2),v_(3),v_(4),v_(5) which represents points in R^(5) listed below and note that A=[[v_(1),v_(2),v_(3),v_(4),v_(5)]] where A is the same matrix as in Part A. v_(1)=[[1],[-2],[0],[0],[2]],v_(2)=[[0],[1],[0],[0],[0]],v_(3)=[[3],[0],[3],[6],[6]],v_(4)=[[-2],[1],[2],[3],[-8]],v_(5)=[[1],[-3],[-1],[-5],[10]] (i) Are there any nontrivial solutions to the vector equation x_(1)v_(1)+x_(2)v_(2)+x_(3)v_(3)+x_(4)v_(4)+x_(5)v_(5)=0? (ii) Let binR^(5) be as in Problem 1. Can you find a unique solution to the vector equation x_(1)v_(1)+x_(2)v_(2)+x_(3)v_(3)+x_(4)v_(4)+x_(5)v_(5)=b ? What is this solution? (iii) Describe Span(v_(1),v_(2),v_(3),v_(4),v_(5)) as a subset of R^(5). (iv) Extra-credit. (3+ points) Describe Span(v_(1),v_(2)) as a subset of R^(5). (v) Extra-credit. (6+ points) Describe the set of solutions to the vector equation x_(1)v_(1)+x_(2)v_(2)=0 in R^(5).
Must complete all parts in part a and part b including the extra credit. please show all the work needed and give a good explanation to each part.
