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In Exercises 23 and $24,$ mark each statement True or False. Justify each answer. a. A homogencous cquation is always consistent.b. The equation $A \mathbf{x}=0$ gives an explicit description of its solution set.c. The homogeneous equation $A \mathbf{x}=0$ has the trivial solution if and only if the equation has at least one free variable.d. The equation $\mathbf{x}=\mathbf{p}+t \mathbf{v}$ describes a line through v parallel to $\mathbf{p}$ . e. The solution set of $A \mathbf{x}=\mathbf{b}$ is the set of all vectors ofthe form $\mathbf{w}=\mathbf{p}+\mathbf{v}_{h},$ where $\mathbf{v}_{h}$ is any solution of the equation $A \mathbf{x}=\mathbf{0} .$

a. Trueb. Falsec. Falsed. Falsee. False

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

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Okay. In this question, what is given five statement here, toe. And what is your question? We have to determine whether each state ministry or files with justifying all of them. Okay, so what? The concept concept is a homogeneous equation. Again. Homogeneous equation off it line passing through P and parallel Truth panel could be in solution set off a X equals to be okay, so we have to justify all five statements, So I will read the statement statement. A homogeneous equation is always consistent. Yes, it is true, because a X equals to zero always has at least one solution. Okay, One solution, and that will be zero solution. Okay, The resolution or we call that travel solution. So this is to you. Okay. On B one of the B B. I read it. The creation X equals to zero. He x equals to zero. Gives an explicit, explicit description off its solution. Said No, it is false. Why? I show you it is not. It will be not explicit. Solving a homogeneous equation gives always on implicit. Okay, implicit description off it. Solution side. So this is falls. Okay, now question. See, we're moving Ocean Sea there it is. Caution. See? And I read it on the homogeneous equation. X equals to zero. Has the travel solution as the travel driver solution? If and only if the equation is at least one free variable. Okay, but and you know it is false. Why? Because because homogeneous equation a X equals to zero has a non rival non travel solution. If only if the question at least one favor. If all will be zero that only the case, it will be travel solution. But if one free one free variable is there and always non driver Okay, then this will be false. On the next we are going to next question T Okay, so what is d d is the equation? X equals toe B plus TV described the line through we The question is a line true P parallel to question is nine through V. They're a little too be. And that is false because it is a line true P and parallel toe way. Okay. And the last e he is the solution. Set off solutions Head off a X equals Toby is the set of all vector off the form W because two p plus V h air. VH is any solution of equation X equals zero. That is also falls because, well, it is an incomplete description off your, um six and assumption is also missing. And this will be true. This will be trained one condition. Then there is exist some vector p such such that a p equals toe Be okay. In this condition, that will be true. But according toa ocean, this is false. So these are over answers e true b false see falls indeed. False and e also false. On this will be our final answer. Thank you.

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