Mark each statement True or False. Justify each answer. a. A homogeneous equation is always consistent. O A. True. A homogenous 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 nontrivial solution. Thus a homogenous equation is always consistent. O B. False. A homogenous 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 homogenous equation is always inconsistent. O C. True. A homogenous 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 homogenous equation is always consistent. O D. False. A homogenous 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 nontrivial solution. Thus a homogenous equation is always inconsistent. b. The equation Ax = 0 gives an explicit description of its solution set. O A. False. Since the equation is solved, Ax = 0 gives an implicit description of its solution set. O B. False. The equation Ax = 0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of OC. True. The equation Ax = 0 gives an explicit description of its solution set. Solving the equation amounts to finding an implicit description of its solution set. solution set. O D. True. Since the equation is solved, Ax = 0 gives an explicit description of the solution set. c. The homogenous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. O A. True. The homogenous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable which implies that the equation has a nontrivial solution. O B. False. The homogeneous equation Ax = 0 always has the trivial solution. OC. False. The homogeneous equation Ax = 0 never has the trivial solution. O D. True. The homogeneous equation Ax = 0 has the trivial solution if and only if the matrix A has a row of zeros which implies the equation has at least one free variable, d. The equation x =p+ tv describes a line through v parallel to p. O A. False. The effect of adding p to v is to move v in a direction parallel to the plane through p and 0. So the equation x = p+ tv describes a plane through p parallel to v O B. True. The effect of adding p to v is to move p in a direction parallel to the line through v and 0. So the equation x =p + tv describes a line through v parallel to p. OC. False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation x =p+ tv describes a line through p parallel to V. O D. False. The effect of adding p to v is to move p in a direction parallel to the plane through v and 0. So the equation x = p + tv describes a plane through v parallel to p. e. 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 equation Ax = 0. b, and there exists a vecto O A. False. The solution set could be empty. The statement is that p is a solution. true when the equation Ax = b is inconsistent for such O B. False. The solution set could be the trivial solution. The statement is only true when the equation Ax = b is inconsistent for some given b, and there exists a vector p such that p is a solution. OC. False. The solution set could be empty. The statement is only true when the equation Ax = b is consistent for some given b, and there exists a vector p such that p is a solution. O D. True. The equation Ax = b is always consistent and there always exists a vector p that is a solution.

## Answer

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a. C is correct
b. A is correct
c. B is correct
d. C is correct
e. C is correct
Solution 3 7X cavation ‘s always consistent As the the triytal solution is always a sowtion 66 AX=0. be at localit but we can not a say there will one non-trivial solution exists. so option 000 000 c is correct rolangan no A is wrong. el 1 ( اوز ) 12 The ecuation Axiao gives an explicit description of itso solution set, It is False as the cauation is soved, Ax=0 gives an implicit descoption of it’s solution set SO option is correct. (BCD) Othere are on the s The homogenous ecuation AX=0 has sowtion if and only if the cavation one free vailable. It is FALSE theee is aluay’s solution for the cavation AXTO is correct. d The cavation X=p+tv describe a line through u paralle to P. as X=PtN PO describe as line goes through and pacalled to v. trivical has at least As a trivial so 2 V (1) 31 32 It is false 17 wape sol, ☺ is correction e the solution set of ax=b is the set of all vectors of the foom waptun, where un is any solution of the eciation AXZOGU 19ə, 13 ② Falle. As the sowtion Set can be empty. And the Ax=b is statement is the only when is corect. consistent for o Some br. d.