Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. ○ A. True. A homogeneous equation can be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax = 0 always has at least one solution, namely x=0. Thus, a homogeneous system of equations can be inconsistent. OB. True. A homogeneous equation cannot be written in the form Ax =0, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax=0 does not have the solution x = 0. Thus, a homogeneous system of equations can be inconsistent. OC. False. A homogeneous equation cannot be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in Rm. Such a system Ax = 0 does not have the solution x = 0. Thus, a homogeneous system of equations cannot be inconsistent. OD. False. A homogeneous equation can be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in R. Such a system Ax = 0 always has at least one solution, namely x=0. Thus, a homogeneous system of equations cannot be inconsistent. b. If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero. Choose the correct answer below. OA. True. A nontrivial solution of Ax = 0 is a nonzero vector x that satisfies Ax = 0. Thus, a nontrivial solution x cannot have any zero entries. B. False. A nontrivial solution of Ax = 0 is a nonzero vector x that satisfies Ax = 0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero. C. True. A nontrivial solution of Ax = 0 is a nonzero vector x that satisfies Ax = 0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero. D. False. A nontrivial solution of Ax = 0 is the zero vector. Thus, a nontrivial solution x must have all zero entries. c. The effect of adding p a vector is to move the vector in a direction parallel to p. Choose the correct answer below. A. False. Given v and p in R2 or R3, the effect of adding p to v is to move v in a direction parallel to the plane through p and 0. B. False. Given v and p in R² or R³, the effect of adding p to v is to move v in a direction parallel to the line through v and 0. OC. False. Given v and p in R² or R³, the effect of adding p to v is to move v in a direction parallel to the plane through v and 0. OD. True. Given v and p in R² or R³, the effect of adding p to v is to move v in a direction parallel to the line through p and 0. d. The equation Ax = b is homogeneous if the zero vector is a solution. Choose the correct answer below. A. False. A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in . If the zero vector is a solution, then b Ax A0=0, which is false. OB. True. A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in R. If the zero vector is a solution, then b = Ax = A0 = 0. OC. False. A system of linear equations is said to be homogeneous if it can be written in the form Ax=b, where A is an mxn matrix and b is a nonzero vector in R. Thus, the zero vector is never a solution of a homogeneous system. OD. True. A system of linear equations is said to be homogeneous if it can be written in the form Ax=b, where A is an mxn matrix and b is a nonzero vector in R". If the zero vector is a solution, then b = 0. e. If Ax = b is consistent, then the solution set of Ax = b is obtained by translating the solution set of Ax = 0. Choose the correct answer below. ○ A. True. Suppose the equation Ax = b is consistent for some given b. Then the solution set of Ax = b is the set of all vectors of the form w=p+v, where v is not a solution of the homogeneous equation Ax = 0. OB. True. Suppose the equation Ax b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w p+v, where V, is any solution of the homogeneous equation Ax = 0. ○ C. False. Suppose the equation Ax = b is consistent for some given b. Then the solution set of Ax = b is the set of all vectors of the form w=p+v, where v₁, is not a solution of the homogeneous equation Ax = 0. OD. False. Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w=p+V, where V, is any solution of the homogeneous equation Ax = 0.