Question

Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, 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 O B. True. A homogeneous equation cannot be written in the form Ax 0, where A is an mxn matrix and O is the zero vector in R. Such a system Ax 0 does not have the sclution x 0. Thus, a homogeneous system of equations can be O C. False. A homogeneous equation cannot be written in the form Ax-0, where A is an mxn matrik and O is the zero vector in , Such a system Ax 0 does not have the solution x 0. Thus, a homogeneous system of equations cannot equations can be inconsistent inconsistent. D. Fa se. A homogeneous equation can be w en n the form Ax-0, where A is an m x n matrix and。เร the zero vector in Such a syster b. If x is a nontrivial solution of Ax 0, then every entry in x is nonzero. Choose the correct answer below. O A. True. A nontrivial solution of Ax o is a nonzero vector x that satisfies Ax 0. Thus, a nontrivial solution x cannot have any zero entries Ax:O always has atleast ne solution, namely r- . us, a homogeneous system of equations cannot be inconsistent. B. Falso. A nontrivial solution o, Ax-Ο is a nonzero vector x that satisfies Ax 0. Thus, a non ri ial solution x can have some zoro en ries so long as not all of its o tries are zero. True. A nontrivial solution o Ax-o is a nonzero vector x hat satisfies Ax=0. Thus, a non rval solution x can have some zero entres so long as not all of its entries are zero. False. A nontrivial solution of Ax = 0 is the zero vector. Thus, a nontrivial solution x must have all zero entries. O c D. c. The effect of adding p to a vector is to move the vector in a direction parallel to p. Choose the correct answer below. OA. 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 O O 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 O C. False. Given v and p in R2or R3, he effect of adding p to v is to move v in a direction parallel to the plane through v and o D. True. Given v and p in R2or R3 the effect of adding p to v is to move v in a direction parallel to the line through p and 10

Answer 1

(a) A homogenous system of equations can be inconsistent. Answer (D) False, A homogeneous equation can be written in the form Ax = 0, where A is an m times n matrix and 0 is the zero vector in R^m such A system Ax = 0 always has at least one solution, namely x = 0. Thus equation can�t be inconsistent. (b) If x is non-trivial solution of Ax = 0, then every entry in x is non-zero Answer False, and Answer is (B) (c) The effect of adding p to a vector is to move the vector in a direction parallel to p Answer (D) True (d) The equation Ax = b is homogenous if the zero vector is a solution. Answer True and Answer is (B) (e) If Ax = b is consistent, then the solution set of Ax = b is obtained by translating the solution set of Ax = 0