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Let {v1, - Vn_ be a spanning set for an inner product space V . Suppose that x and y are such that (x, Vk) = (y, Vk) for all k Prove that x = y_...

Question

Let {v1, - Vn_ be a spanning set for an inner product space V . Suppose that x and y are such that (x, Vk) = (y, Vk) for all k Prove that x = y_

Let {v1, - Vn_ be a spanning set for an inner product space V . Suppose that x and y are such that (x, Vk) = (y, Vk) for all k Prove that x = y_



Answers

Let $V$ be an inner product space with basis $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\} .$ If $\mathbf{x}$ and $\mathbf{y}$ are vectors in $V$ such that $\left\langle\mathbf{x}, \mathbf{v}_{i}\right\rangle=\left\langle\mathbf{y}, \mathbf{v}_{i}\right\rangle$ for each $i=1,2, \ldots, n,$ prove that $\mathbf{x}=\mathbf{y}$.

In this problem. We have an Ortho agonal set of vectors, and we're told that these vectors are all orthogonal. And that's how we know that there are orthogonal, um, to make them Ortho normal. We have to also make each of the vectors normal. And so, um, so if we take any of the vectors v i and we divide it by the length of the I, well, that will have the same direction as V I, but it will be, um, normal. And so that's four. Oh, I from one tu que So now our normal set would just be Ortho normal set because they're already orthogonal. And to make them normal, we just have to do that divide by their length. Um, then it's just you won. You, too. Dot dot dot. You okay?

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'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'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'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'm going to use property to to rewrite all of these as VW. So it'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'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'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's not the same. So the normal V plus w squared is I ve thus double you V plus double use. That'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'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's V minus w So race a little bit here was thinking that'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'm not sure that's right. So let me think about what's going on here with this last one. It'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's going to be double you here, and let'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's rate these out. Let'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've got positive w So we've got two double use also, So that'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's left and I'm going to rewrite them as, ah, VW, the Enterprise V and W according to property, too. Okay, plus minus plus minus, there's cancel out, and the inner product of V and V is the norm square. So it's too times norm of V squared, plus norm of W. Squared, and that's what we wanted to prove.

I'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's property to this equals that this that's gonna be property three and then zero times anything is zero.

We have to show that the norm of V plus w squared is the norm of V plus two times the inner product of V and W plus the norm of double you squared. Well, I'm thinking that that is just the norm of V and V plus two, norm of V and W plus the norm of double you and double you. Okay, so a hint. And I'm gonna write down the hint here. The norm of V plus double you squared is just the plus double you the inner product of V plus W and V plus w Whoops. Okay, that makes sense. Based on what I just did up here, Um, because if you're going to get the norm of V plus W, then according to the definition, you would have to take V plus W the inner product of V plus W and V Plus doubled w and square it. No, no. And then take the square root. So squaring both sides, you get what's written above. Okay, But then weaken. Separate that into the V plus the inner product of V and W, plus the inner product of W and V plus the inner product of W and W. These are all supposed to have vector symbols on top of them. And so we can see that these two are gonna end up being to V. W. And so that leads us to hear which leads us back to their maybe two vectors V and w. Okay, so four orthogonal in real space, that would be 90 degrees factors, um, be the inner product of V and W is zero. So if we have V. W, let's just say that this is the length of V and this is the length of W. Then this would be the length of V plus W. And we just learned that the plus w is the norm of V squared groups, plus two times the inner product, plus then norm of W squared. But if we have a 90 degree angle than this, become zero and in our this is they're supposed to be a squared over here. Yeah, sido this is the same as see in our drawing this would be the same as a and this would be the same as being are drawing. So C squared equals a squared plus B squared


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