use-the-inner-product-axioms-and-other-results-of-this-section-to-verify-the-statements-in-exercis-4

Question

Use the inner product axioms and other results of this section to verify the statements in Exercises $15-18$ .
$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

Angelo Rendina · Accepted Answer

let&#x27;s combine in just one computation, both the plus and minus case. So the length of you plus or minus V squared. So this is you plus or a minus V against you, plus or minus B. But with these notation, we mean that we either always take the blasts. All we always take the miners now is the linearity in the first argument that&#x27;s you against you. Plus or minus V plus or minus V against you, plus or minus B. Now we&#x27;re still in a charity. In the second argument, we got you against you plus or minus you against V Blaster miners. And now we put in brackets the whole second in their product. So we have V against you blossom minus V against me. Now observe that you against you is just the length of your squared. And then we have plus reminders twice you ve where the two comes from. The fact that both you and we and we and you are the same because of the symmetry. And we&#x27;re something them with the same sign. So it&#x27;s other two pluses or two minuses. So it&#x27;s plus or minus two you ve and then the last term is a plus in green because he re multiplying pastor minors by plus or minus so either that both plus or the boat minus. When we multiply to turn together, you always get plus, So the remaining term is plus V against Visa that the length of B squared an hour done. We just write the sum. And so the plus reminder simplifies out zero, and you have twice the length of you squared, plus twice the length of the square as we wanted.

Angelo Rendina · Answer

you and we form a more formal basis for the vectors pays every considering we want to study the lamps off humanity. So instead of considering the length of your ministry, that&#x27;s considered square. And we know that disease the inner product off humanist be against itself against humanity. Now use the linearity in the first argument so that you against humanity, then minus V against humanity. And now again, we used dealing. Art is in the second argument. So listen, old Wheat in right, the first in the product and we green the second in a product. So for the first in a product and right, we have you against you, minus you against the then we have minus now in green for the second product V against you. And then we have minus minus. That becomes a plus V against V. Now you against you is one. Because remember you Visa Ortho. Normal basis. So you&#x27;re against you is one that the normal part you against V zero. That&#x27;s the orthogonal part Be against you. Similarly, zero. And we against V is one. It&#x27;s all in all this is too. And now we&#x27;re done because the land of you minus V is just the root of these sorrow too

Donald Albin · Answer

So I need to prove that you the inner product of you with V plus double you is the inner product of U and V plus the inner product of you and W. Okay, so this is very much like, um, Property three, which says that, um So I&#x27;m gonna write property three. I am going to write V plus W comma. You equals you w plus V w. Okay, well, this is the same as you, V plus W because of property, too. So Property three is here, okay? And one property, too is here. And then we can use just, um this is supposed to say you up here. Oh, my goodness is supposed to say that&#x27;s re do this. I think that is very wrong. Um, you double you. All right? I wrote this completely wrong. It&#x27;s supposed to save you and w you all right. So let&#x27;s let&#x27;s keep going. Um, Then we can use property to to just switch these two around. So that&#x27;s going to give us you the plus. You w Okay. So, again, you could write the&#x27;s just like last time in a different order, probably starting here and then writing that and then writing that and then writing that

Donald Albin · Answer

I&#x27;m going to prove that the inner product of V and the zero vector is zero. Well, first of all, the zero vector is the same as zero times any vector. And so I can write that one vector times. Another vector is the same as this vector times. This vector that is using, um, property, too. We can reverse the order. Um, then using property three using property three. This is zero times you V, which is zero. And this is V zero. And so if you write this, you might want to start here and proceed in this direction. So what I see here is just using a definition. Then using this equals that that&#x27;s property to this equals that this that&#x27;s gonna be property three and then zero times anything is zero.

Donald Albin · Answer

all right now we&#x27;re going to figure out what the inner product of you plus V and W plus X would be well, first of all, according to Property four, that&#x27;s the same as you w plus X plus V w plus X And then according to what we did in Problem 30 that&#x27;s the same as you w plus you, the inner product of you and X plus the inner product of V and W plus the inner product of the and X.

Donald Albin · Answer

So first we need to prove that, um uh, looking for my paper here. Okay, this says ah v plus W squared minus the minus w squared equals four v double you interesting. So the norm of V plus w squared would be, um, see here who would be the inner product of V Israel vectors plus double you and the plus w Okay. And of course, the norm of the minus w squared is going to be V minus double you and V minus double you. So let&#x27;s write this out. The inner product of V plus W and V plus W minus the inner product of the minus w and V minus W That&#x27;s just using the definition of the norm. Now, according Teoh, Um, Property four. We can make this a the V plus W plus w the plus w minus. The whoops. Got my symbols in the wrong place. Okay, so V V minus double you minus negative. W the minus w Which, of course we could using property three. Factor out the negative one, so that gives us positive. Okay. Using property to weaken, switch these around and now, using property four again, we can separate them. V the plus W v plus B w plus w double you minus V the minus negative w the plus. So I&#x27;m running out of room. Gonna have to get the next line V double you plus negative double you double you. And then according to Property Three, we can put the negative outside here and we can change these two positive Okay. V v minus vv w w minus w w. And I&#x27;m going to use property to to rewrite all of these as VW. So it&#x27;s V W the inner product of V N W plus the inner product of V A and W plus the inner product of V and W plus the inner product of E and W. And there are four of them and that&#x27;s what we intended to prove be Now we need to show that the norm of V plus w squared plus V minus double you squared. Well, that&#x27;s the same thing for you. Plus w squared The minus w squared is two times the norm of V squared plus the norm of double You squared. Okay, so this is actually the same. Oh, no, we got a plus in between you. This is Gata minus in between. And then this is gonna plus in between. All right, so it&#x27;s not the same. So the normal V plus w squared is I ve thus double you V plus double use. That&#x27;s the inner product of E plus folk w and V plus W plus the inner product of V plus W the plus w. That&#x27;s the same as the inner product of V and V plus W plus the inner product of W and V plus W plus the inner product of V. Um, the W no, that&#x27;s V minus w So race a little bit here was thinking that&#x27;s the exact same thing. Necessarily make sense. So these were minus It is okay, so the second part is V minus W All right, so that was still part of the first part. Okay. Plus V comma, the minus double. You plus negative v comma V minus. W. All right. Um, we can factor that negative out using property three. And now we can write him in reverse according to property, too. All right. No. Separate him again. V v Plus in her product of w and the plus the inner product of the and W plus the inner product of W and W, plus the inner product of V and the plus the inner product of Negative W and V. Morning of space minus the inner product. Interesting. I&#x27;m not sure that&#x27;s right. So let me think about what&#x27;s going on here with this last one. It&#x27;s, um v V minus w plus V V minus w oh V V minus W This should be double you V minus W So that&#x27;s going to be double you here, and let&#x27;s see race on, uh, race this stuff. Okay. 12345 cases. This was going to be negative. Double you v minus the W minus. Negative double. You double you. That looks better. All right, so let&#x27;s rate these out. Let&#x27;s see how many VVS we have. We got one there on one there, and, um, this one down here factor the negative. W out according to property three, and we&#x27;ve got positive w So we&#x27;ve got two double use also, So that&#x27;s going to be too of the V Weise Inter product V and V Plus two whoops area affected out the to the inner product of double you and W. And then we have all the stuff that&#x27;s left and I&#x27;m going to rewrite them as, ah, VW, the Enterprise V and W according to property, too. Okay, plus minus plus minus, there&#x27;s cancel out, and the inner product of V and V is the norm square. So it&#x27;s too times norm of V squared, plus norm of W. Squared, and that&#x27;s what we wanted to prove.

Problem

Given $a \geq 0$ and $b \geq 0,$ let $\mathbf{u}=…

Problem 18 Medium Difficulty

Use the inner product axioms and other results of this section to verify the statements in Exercises $15-18$ .
$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

Answer

$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

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

Linear Algebra and Its Applications

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Kristen K.

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$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

Linear Algebra and Its Applications

Catherine R.

Kayleah T.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Catherine R.

Kayleah T.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications