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In Exercises 23 and $24,$ mark each statement True or False. Justify each answer. a. If $\mathbf{x}$ is a nontrivial solution of $A \mathbf{x}=\mathbf{0},$ then every entry in $\mathbf{x}$ is nonzero.b. The equation $\mathbf{x}=x_{2} \mathbf{u}+x_{3} \mathbf{v}$ , with $x_{2}$ and $x_{3}$ free (and neither $\mathbf{u}$ nor $\mathbf{v}$ a multiple of the other), describes a plane through the origin.c. The equation $A \mathbf{x}=\mathbf{b}$ is homogeneous if the zero vectoris a solution. d. The effect of adding $\mathbf{p}$ to a vector is to move the vector in a direction parallel to $\mathbf{p}$ .

a. False.b. Truec. Trued. Truee. False

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

Missouri State University

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

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Okay, So this problem, we're going to check whether the following statements are true or false. So the first statement says, um, if axis a nontrivial solution up off X equals zero than every entry in access none. Zero. So this is clearly false because we can consider a solution. Consider such a solution that it's x three one and zero x three. So in this case, ex three's a pre bearable. But, um, the specter asked, contains zero, which is the second entry here. So what? That means our our first aim in this wrong. So that's we will mark falls here. All right, the second statement, Now we have ext because x two times you us x three times fee. So we need to check that, um, whether this is the plane that that is passing through the origin. So now remember how we describe a plane as how we describe a lying in ah are to space. So that is acts equals you times you plus tee times feet where you and V are vectors and tease is ah, real number. So similarly, we can express playing in our three space. That is you times as a plus B plus the ice tea class W your W ys accounts tend the casting term, and we have two variables here that determines the plane playing in the, uh that determines the plane are three space. But right now, the key thing here is the casting term. W now recall that the constant, constant terms the cats in terming are two space that represents the The point is that the line is passing through, and so in the arteries are three spacing the plane. So if we have the constant that is zero, that means that means that this plane is passing through the already origin. So that means in this case, we have x two times you and x x three times V. But we don't have any. We don't have any constant terms. So that means the council terms is the reflector. So that that implies this plane is passing through the origin. So we're done. All right, this third a statement. So excess the equation. There he acts equal speeds homogeneous. See if the zero vector is a solution. No. Now recall the detonation of homogeneous equation. That means, uh, we have a homogeneous equation. So That means the baby has to be zero because we don't allow any consented. None zero constant. On the right hand side, we have a homogeneous equation, so as chatty If this is true, so zero is a solution. So that means we plugging zero. So a times zero vector will be zero. And that is exactly be, because no matter better what the matter was solution. We put in our equation in a basket this council be, but, uh, here we put in zero, so it must be zero or so. So that implies he is a zero constant. And so and so we have a tax equals zero. So this is a homogeneous system, and we're done. All right, The fourth statement. So it says, um effect off Adam P two a vector is to move the with the factory in a direction parallel to Pete. No. Oh, before we start 1/4 1 out just right. Right down. True or force here. So 2nd 1 is true? Yeah, the first ones Paul's 2nd 131 are all true. So the 4th 1 So right now here we're given I say we're given a vector. Say, b how Drove it in pictures. Um, I'm trying to be this way. So we're given the vector the and also we are adding p to this vector. Me. So let's say P is, uh is this this is Pete. So to perform the vector edition withdrawal such a thing here. And finally we will get this and this is the plus peak. And we can we can see from a picture that we're moving be from this point directly to this point. And it is along the direction off p. We're moving straightly, right? Poor. My word in this direction the direction of P. And so that means it is It is exactly just like a translation here. So it's true.

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