let-a-be-an-m-times-n-matrix-and-let-mathbfu-and-mathbfv-be-vectors-in-mathbbrn-with-the-property-th

Question

Let $A$ be an $m 	imes n$ matrix, and let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^{n}$ with the property that $A \mathbf{u}=\mathbf{0}$ and $A \mathbf{v}=\mathbf{0} .$ Explain why $A(\mathbf{u}+\mathbf{v})$ must be the zero vector. Then explain why $A(c \mathbf{u}+d \mathbf{v})=\mathbf{0}$ for each pair of scalars $c$ and $d$

Runpeng Li · Accepted Answer

All right. So in this problem, work even eight times you equals zero and the times be it was zero. Now, first we need to figure out eight times. You plus B. Now, this is quite simple. We just We know that a matrix matrix multiplication is distributed so unjust that apply the law, which is eight times you plus 800 now. It had serious 08 times v zero. Also, it is zero now. Secondly, we need Thio. Find out many times. See Time&#x27;s Hugh plus the Times V Now C and D are skaters. So again we apply the, um it&#x27;s reputed ity. So that&#x27;s 80 times. See, do you plus eight times Dean. Yeah. Now, since Cmdr scale er&#x27;s, we can take the skater impromptu off the expression so that it&#x27;s C times. Hey, you last e kinds a V. Now, right now, I can apply this associative ity off matrix multiplication so that it&#x27;s C times zero us d times zero. So that is zero and

Michael Jacobsen · Answer

In this example we&#x27;re dealing with the M by N Matrix that satisfies the homogeneous equation. A X equals zero. Let&#x27;s for the sake of an interesting problem. Suppose that the vector you in R N has to always match this size. Here is a solution. Let&#x27;s say what that means for the moment. If you in our in is a solution, that would mean that a times you where you was in place of X is the zero vector. Next, let&#x27;s further do the following let see be any scaler and we want to know then is see Time&#x27;s you a solution to the same matrix equation eight times X equals zero. So given that you is a solution, we take a scaler, see, And we&#x27;re wondering is see time&#x27;s you also a solution? The only way to check and know for sure is to place our c times you here in place of X. So here&#x27;s our check. Take the matrix A Multiply it by the vector see times you. Now our goal is to make it to the zero vector here. If we can&#x27;t get there, will say no See times, you is not a solution. If we make it to this euro vector, they will say yes. See, times you is a solution. So let&#x27;s see how to work out our equation down here So far. Well, one thing we can do is use the community Tiv ity of matrix vector multiplication that allows a scaler to commute to the outside to the left of the matrix. A. So a time See you is equivalent to scaler multiplying eight times the vector you. But then we can say you is a solution to them matrix equation. With that information, this would turn into C times the zero vector, since a U equals zero is what it means to have a solution. But sometimes a zero vector is always equal to the zero vector, no matter which scaler we pick. So this shows that C times you is also a solution. Now we also know that C was any real number. So this tells us as soon as we find one solution you you can scale that particular vector as much as you like with any positive or negative number C times you would still be a solution

Jason Li · Answer

you want to prove you plus negative u u zero where you was a two d vector. So that&#x27;s a you his equal toe a B for arbor cherry. The old numbers baby notice that negative one times you It&#x27;s equal to negative one times a a ticket of one times, please. And this is equal to negative. Thank you. So therefore, negative, you busy quarter. I think it is a maybe to be. But now we have you first. Negative. You is equal to a B, but they have a negative be substituting you here and they did of you. Now, using our definition is a vector addition. We have a plus making TV and you negative because Spector Edition is component wise and this is equal to zero zero since a NPR riel numbers and we&#x27;re adding their additive imprints. But this back to 00 has also returned at zero where this is the zero factor, not zero rial number. So we have now proven that you plus negative u equals zero

Abdul Vahid M · Answer

here. The you like this given us gave me on be able to show moving helpless zero no does say is a victim and denoted by better symbol that one you have to show because equal you look, you hear. Okay, so the idea is the concept of a young member tradition. So you bless your heart. Wait. 03 Great. You don&#x27;t want to write this, right? So when you add left hand side of your taking up defensive but issue princey book like that&#x27;s what is your But I think it was 00 taking us. So just specter traditional here. And, you know, when you add any number, you get the same number because the roysten similarly being again actually. What what constantly use beauties. This heading. I decided, you know, we started okay.

Abdul Vahid M · Answer

been giving you Victor. You can understand the same big again. Take a moment. We can wait first later yet plus See you. You. This is you. Did you since you? Because he couldn&#x27;t care. Thanks. The thing is one of the no this a this digit zero pants. Yes, Yes. You know you get started lettuce and that&#x27;s assists U S units. Thanks. A STD dont creates any number. Are you going to beat it against you? This actually what? That&#x27;s we are care times That is a good care times its development C&#x27;mon depend on his jailer because you have B a d It is You look so it is similar to the reputation will be doing created since in Riyadh on boats, people, any number I think you visited independence against you sending out baptisms get zero victory. Did you

Doruk Isik · Answer

in this problem, we&#x27;re given a set of bases and we know the vectors that formed this place is so you know to be and where has to explain why actors are, uh this spaces are the collectors off the on by an identity make. So if this bees are the basis in vector space, be damn You know, that director can be written as a linear combination off these basis vectors. And this it&#x27;s true before device inspector itself. So we can like be one as one time one plus zero b two plus dr 0.0 b m. So don&#x27;t be too would be zero times be one plus one time to be too Plus zero times bien I&#x27;m gonna be an would be zero b one plus zero view too So old zeros up to be end up Enter Amanda and the answer would be so from this, we can find the coordinated vector for anyone to be one old zeros. And from the second equation, be fine reporter factor for me to to be 010 hold zeros and from 30 question we find the court inspector for intern and director to be 000 old zeros. Why intern? So last term is one says you get sick. Or did that reporter? First time is the first column off the identity matrix for a second when it is the second Poland A or B M for it is defense co

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Alayna H.

Kayleah T.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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Linear Algebra and Its Applications

Alayna H.

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Absolute Value - Example 1

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Alayna H.

Kayleah T.

Kristen K.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications