Mark each statement True or False Justify each answer: a. A homogeneous equation is always consistent OA False. A homogenous equation can be written in the form Ax = 0, where A is an mxn matrix and is the zero vector in Rm Such system Ax = 0 always has at least one nontrivial solution Thus homogenous equation is always inconsistent: 0 B. True homogenous equation can be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in Im Such system Ax = 0 always has at least one nontrivial solution Thus = homogenous equation is always consistent. False. homogenous equation can be written in the form Ax = 0, where A is an mx matrix and is the zero vector in Rm Such system Ax = 0 always has at least one solution; namely; x = 0. Thus homogenous equation is always inconsistent: True A homogenous equation can be written in the form Ax = 0, where A is an mxn matrix and is the zero vector in Rm . Such system Ax = 0 always has at least one solution; namely; x =0. Thus homogenous equation is always consistent: The equation Ax = 0 gives an explicit description of its solution set: OA True The equation Ax = gives an explicit description of its solution set. Solving the equation amounts to finding an implicit description of its solution set: False. Since the equation is solved, Ax = 0 gives an implicit description of its solution set: True Since the equation is solved, Ax = 0 gives an explicit description of the solution set: False. The equation Ax = gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set: The homogenous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. False. The homogeneous equation Ax = 0 always has the trivial solution. True The homogeneous equation Ax = 0 has the trivial solution if and only if the matrix _ has row of zeros which implies the equation has at least one free variable False. The homogeneous equation Ax = 0 never has the trivia solution. True The homogenous equation Ax = has the trivial solution if and only if the equation has at least one free variable which implies that the equation has nontrivial solution_ d. The equation X = p tv describes line through parallel to p_ True The effect of adding p to is to move p in a direction parallel to the line through and 0. So the equation x = p + tv describes line through parallel to p_ False. The effect of adding P to is to move p in a direction paralle to the plane through and 0. So the equation x = p tv describes plane through parallel to p_ False. The effect of adding P to is to move in a direction parallel to the line through p and 0. So the equation x = p describes line through p parallel to False. The effect of adding p to is to move in a direction parallel t0 the plane through p and 0. So the equation x = p = tv describes plane through p parallel to
The Correct Answer and Explanation is:
Let’s go through each statement one by one and analyze them:
Statement a: “A homogeneous equation is always consistent.”
- Answer: True.
- Explanation: A homogeneous system of linear equations is always consistent because the system can always be written in the form Ax=0A \mathbf{x} = 0Ax=0, where AAA is an m×nm \times nm×n matrix and x\mathbf{x}x is the vector of variables. The system will always have at least the trivial solution, x=0\mathbf{x} = 0x=0. In other words, the system Ax=0A \mathbf{x} = 0Ax=0 always has at least one solution, and that solution is x=0\mathbf{x} = 0x=0, making the system consistent. There is no scenario where a homogeneous system is inconsistent, as it will always have at least the trivial solution.
Statement b: “The equation Ax=0A \mathbf{x} = 0Ax=0 gives an explicit description of its solution set.”
- Answer: False.
- Explanation: The equation Ax=0A \mathbf{x} = 0Ax=0 does not provide an explicit description of the solution set until the system is solved. When the system is solved, you obtain either the trivial solution or the general solution (which could include free variables if the system has infinitely many solutions). Thus, Ax=0A \mathbf{x} = 0Ax=0 is an implicit description, and solving it provides an explicit description of the solution set.
Statement c: “The homogeneous equation Ax=0A \mathbf{x} = 0Ax=0 has the trivial solution if and only if the equation has at least one free variable.”
- Answer: False.
- Explanation: The homogeneous system Ax=0A \mathbf{x} = 0Ax=0 always has the trivial solution x=0\mathbf{x} = 0x=0, regardless of whether there are free variables or not. The existence of free variables in the solution corresponds to the system having nontrivial solutions, but the trivial solution always exists in a homogeneous system. The statement is incorrect in suggesting that the trivial solution depends on the presence of free variables.
Statement d: “The equation x=p+tv\mathbf{x} = \mathbf{p} + t \mathbf{v}x=p+tv describes a line through p\mathbf{p}p parallel to v\mathbf{v}v.”
- Answer: True.
- Explanation: The equation x=p+tv\mathbf{x} = \mathbf{p} + t \mathbf{v}x=p+tv represents a parametric equation for a line. The vector v\mathbf{v}v determines the direction of the line, and ttt is the parameter that can take any real value. The point p\mathbf{p}p is a point on the line, and as ttt varies, the position vector x\mathbf{x}x traces out all the points on the line. Therefore, this equation describes a line through the point p\mathbf{p}p, parallel to the vector v\mathbf{v}v.
Summary of answers:
- a. True.
- b. False.
- c. False.
- d. True.
