#7 n. Let b e R"(d) п — 5, r — 4that N(A) 0andsuppose1(a) What can you conclude about the column vec-tors of A? Are they linearly independent? Dothey span R"? Explain(b) How many solutions...

#7

Extracted text: 1 2 (d) A = 1 1 2 b= 2 1 1 0 2 1 (е) А — 0 5 1 b= 2 1 2 1 5 2 4 (f) А — 10 b= 1 2 5 5. For each consistent system in Exercise 4, determ- ine whether there will be one or infinitely many solutions by examining the column vectors of the coefficient matrix A 6. How many solutions will the linear system Ax = b have if b is in the column space of A and the column vectors of A are linearly dependent? Explain 7. Let A be a 6 x n matrix of rank r and let b be a vec- tor in R6. For each choice ofr and n that follows, indicate the possibilities as to the number of solu- tions one could have for the linear system Ax = b Explain your answers (a) n7, r 5 (b) n 7, r 6 п (с) п 3 5, r 3D5 8. Let A be an m x n matrix with m > n. Let b e R" (d) п — 5, r — 4 that N(A) 0 and suppose 1 (a) What can you conclude about the column vec- tors of A? Are they linearly independent? Do they span R"? Explain (b) How many solutions will the system Ax = b have if b is not in the column space of A? How many solutions will there be if b is in the column space of A? Explain 9. Let A and B be 6 x 5 matrices. If dim N(A) = 2, what is the rank of A? If the rank of B is 4, what is the dimension of N(B)? 10. Let A be an m x n matrix whose rank is equal to n. If Ac Ad, does this imply that c must be equal to d? What if the rank of A is less than n? Explain