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Let $A$ be any invertible $n \times n$ matrix. Show that for $\mathbf{u}, \mathbf{v}$ in $\mathbb{R}^{n},$ the formula $\langle\mathbf{u}, \mathbf{v}\rangle=(A \mathbf{u}) \cdot(A \mathbf{v})=(A \mathbf{u})^{T}(A \mathbf{v})$ defines an inner product on $\mathbb{R}^{n} .$

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Calculus 3

Chapter 6

Orthogonality and Least Square

Section 7

Inner Product Spaces

Vectors

Johns Hopkins University

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Hello there. Okay, So for this exercise, let's consider first we got a is going to be an end times and matrix in vertebral metrics. Okay? In vertical matrix. Also, we got vectors called you on V on our n. Well, there, Andi, let us define on the inner product for this space as follows. So the inner product Yeah, uh, is defined as taking a you the product with a the Okay, but this is just taking a You just pose and the Okay, So this is how is defined in the product. And we need to show that this is a corresponds to again. I mean, on in their problems. Okay, so we need thio to show that satisfy the actions off animal problem. So let's start the first that we need that we need todo to show. Okay. Is that you the Is it supposed to be used? It means that the order off the inner product off the vectors doesn't affect. Okay, so let's do that. Let's start. So you the is equals to take into the by definition off the product A You proposed a the beef. Then we can expand this oppression here, so this becomes u transpose a transpose a on the Okay. So Well, we know that this term here is going to be just a number so we can take the transpose of this expression on. We're going toe the same value because this here is going to be in a scaler after evaluating after performing it operation between these between the doctors on the matrices. So this is just a number. So if we take here, the transpose of the expression doesn't affect their soul. So when we take here the trust pose we obtained retrace both a Just pose a on you. Okay, so this part here, we can associate these thes terms on what we obtain is that this part is equals to a V transpose on this part here is just a you which corresponds to taking the you. So we have shown the first action. Let's continue the other property they will need to show is that Yeah, the second one. Okay, that we need to show Is that given u plus v taking the inner product off? This is equivalent to taking thio distribute distribution. Okay. The stoop activity property to this plus the the Okay, so we need to show this right. Let's start by taking the left hand side So the left hand side is equals to take in a times you lost the transpose a w Yeah, This is equals to you love v transpose aid to expose a love you And here we can distributed supposed to each component So this is you transpose plus a transpose 88? No. Yeah. Okay, so here we can distribute the operation off A So we obtained u transpose a transpose waas re transposed a transpose times a love you And this is just UT 88 The view plus Bt 80 a love you this part here we can Well, we can rewrite them. Andi, they become a u A fellow of you Waas a Sorry a V here. Just both here. Just both transpose Hey, value on this is just you w class I'm sorry. Here the w So the second action is hold a swell. Let's continue the third one say that if we multiply by any scaler seat, the inner product is equal to take him to see out off the inner product. See you under where c is just like a real member just on the scaler body. Yeah. Okay, so let's show this. So let's consider this part. He you love you is a time. See you transpose a lot of you is just taking C See you transpose a transposed a love you here the sea We can take it out because it's just a killer value So we'll see Hugh transposed a transpose a above you We can regret this and get things See that multiplies a you transpose a love you And here is clear that it is equal to see you enough so we can take out any scale. So this is also hold on. The final action that we need to show is that the inner product off a vector with itself is greater or equal to zero on the equality holds if and only if you is equal to zero. Okay, so let's start. But so in this case, we're going to define a Yeah, you transpose as a one eight to the A n. Okay, so define it in this way. When we compute this you in the product here you think a you transpose a u But this off the evaluating the This is a product we obtain that is equals toe a square plus a two square plus the plus eight and square. So here, clearly each of these components are going to be either zero or greater than zero, because off the square, they're all positive numbers. So these person here is going to be greater or equal than zero. So the first part is checked. Now, the second one, when this is equals to zero, so you is equal to zero. This is equivalent to say that a one square plus a two square plus that a N square is equal to zero. Okay, but this is equivalent to say that each of these components the only possibility to this estimation off Bostic numbers to be equals to zero is that all the elements are zero. So that implies that a one a two is equal to the goes to a n on a lot. These terms are zero. Okay, if all these terms are zero, that means just by what we have to find that a U is equal to zero. But we got that a is an in vertebral matrix. So this implies this is equivalent. To say that you ease equals must be equals to zero. So we have shown that the inner product off you with itself is equals to zero if and only if you is equal to zero and that's it, that's all. So we have shown that the A yeah, that for any Haymon's A matrix A. And in vertical metrics, I can the inner product off two vectors you can be on our n define as a u transpose a the peace I'm in their product.

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