Problem 6. Determine whether each of the following statements is true or false. Provide a justification for each of your responses.
(a) If the coefficient matrix of a linear system has a pivot in the rightmost column, then the system is inconsistent.
(b) If a linear system has two equations and four unknowns, then it must be consistent.
(c) If a linear system having four equations and three unknowns is consistent, then the solution is unique.
(d) Suppose that a linear system has four equations and four unknowns and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.
(e) Suppose that a linear system has five equations and three unknowns and that the coefficient matrix has a pivot in every column. Then the linear system is consistent and has a unique solution.
Problem 7. Determine whether each of the following statements is true or false. Provide a justification for each of your responses.
(a) Given two vectors v and w, the vector 0 is a linear combination of v and w.
(b) Suppose that v1, v2, ..., vn is a collection of m-dimensional vectors and that the matrix [ v1 v2 ... vn ] has a pivot position in every row. Then every m-dimensional vector b can be written as a linear combination of v1, v2, ..., vn.
(c) Suppose that v1, v2, ..., vn is a collection of m-dimensional vectors and that the matrix [ v1 v2 ... vn ] has a pivot position in every row and every column. Then every m-dimensional vector b can be written as a linear combination of v1, v2, ..., vn in exactly one way.
(d) It is possible to find three 4-dimensional vectors v1, v2 and v3 such that every 4-dimensional vector can be written as a linear combination of v1, v2 and v3.
Problem 8. Suppose that
A = [ 3 -27 ; -2 18 ].
(a) Describe the solution space to the equation Ax = 0.
(b) Find a 2 x 3 matrix B with no zero entries such that AB = 0.