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2) Let A be an n X non-singular matix. Show that (v, w) oT AT Aw defines an inner product on R" When is this the SAme as the standard dot product? th hontom Of...

Question

2) Let A be an n X non-singular matix. Show that (v, w) oT AT Aw defines an inner product on R" When is this the SAme as the standard dot product? th hontom Of Lecture rond &66 Corollarv %1

2) Let A be an n X non-singular matix. Show that (v, w) oT AT Aw defines an inner product on R" When is this the SAme as the standard dot product? th hontom Of Lecture rond &66 Corollarv %1



Answers

Let $V$ be the vector space of $m \times n$ matrices over $\mathbf{R}$. Show that $\langle A, B\rangle=\operatorname{tr}\left(B^{T} A\right)$ defines an inner product in $V$.

Hello there. So for this exercise we're going to consider any inner product space where W. V. One and V two are elements of the vector space V. So, um we need to show that if the view is orthogonal will be one and V two, then Dove is also familiar with a linear combination of these two factors that is greeting here. So how to show this? Well, the meaning of Ortho? No. Okay. Is that these vectors the inner product of W. We B one Is equal to zero. And also that the inner product of the view would be to Equals zero. So what we need to show is that it didn't satisfy. Then that implies that I love you in their product with factor K one, B one plus K two. The two is also equal to zero. So for that we're going to use the properties of the inner product. So let's remember that the inner product is linear. For real vector spaces is linear for each of the each of the components of the problems. That means that we can write this as The younger brother of W. with K one. B 1 plus the inner product with W. With K two, the two. And then this K. one and K two are just real numbers. So K. one and K two are just real numbers. So we can take out of the product because of the um virginity axiom. So this becomes Okay one. The inner product of the view with K one plus gate two. Inner product of I love you with kate with Me too. Here's a. b. one and we know that these elements are equals to zero by the hypothesis here. So we know that W is orthogonal would be one and V. Two. That means that These two in their products are equals to zero and that result as zero. So we have shown that the view being the product of top with K one, B one plus K to read too Is equal to zero.

They're so in this occasion we're going to consider the space of polynomial of degree two. With the revelation in their product. That means that we have a set of points where we are going to ever eat our point annuals. And what we're going to do is taking the sum of the multiplication of these two point almost evaluated at those points. So in this case the points that we are going to have a wider point annuals are zero equals 2 -2. The zero other point to. And these are the point of us that we have here. So what we need to show is that these two point terminals are orthogonal, which is equivalent that the inner product Is equal to zero. So let's take these let's calculate this uh in the product. So then they're proud of B Q evaluated at the point x zero x one x two. It's going to be equal to be x zero cube exit europe plus P x one Q x one loss P x two, cute X two. Then Let's avoid the points. So we got here. Extras -2. So be evaluated at -2 is just -2 times minus two square for the point x one. That is equal to zero. Both point animals will be equal to zero, so it's just 10 And the last term is To hear so it's two times 2 square. You can observe that this is equal to minus eight plus eight, which clearly is equal to zero. Therefore these two point normals are orthogonal.

Hello there. So in this occasion we got the space of polynomial of degree two. And with the inner product defined as the integral between -1 and one of the notification of the plane annuals. So uh given this space within their product, we need to determine if the point MLP and Q. R north or no. So remember that the notion of or tonality is given by the inner products. So if the inner product of those two vectors is equal to zero, then we can say that P and Q are are a phone. So this is what we need to verify in this case. And one important remark is that your functionality depends on the notion of the problem. So we can have the same space with a different weird in their product and then the Arthur finality will completely change. Okay, so just to give that remark, let's let's start to calculate. So we need to calculate the inner product of P with Q. That is equal to the integral between -1 and one of minus one minus X. Lost two X squared times two x plus x is square the X. Okay If we expanded expression we obtain from 1 to 1 two x 2. The four was three x cube -3 x square and -2 x. The X. So in this case when you you have this kind of symmetric intervals geometrically, that means that you are integrating between -1 and what and when you have this kind of integral, if you have an odd function then the integral will cancel out. That means that that integral suppose that this corresponds to some function F. That is off. Then the integral between that symmetric interval in this case -11. But it could be any symmetric interval minus a. If f is off then this will be zero and in case that you have and even function. So in case that you have a gene that is even then the integral will produce 22 times the integral between zero and a. Of that function. The X. Why? Because he's enough to give have the interval of this part and then multiply by two because the part here is completely safe. So that's a really important property that happened with functions and you have always to keep in mind because it will help you to reduce a lot of calculations. So in this case you can say that in this case this memorial is off so the interview will be zero and for this millennial is the same. So you and with two even functions that you need to integrate. So this will be two times the integral between zero and one of two X. To the four minus three X square the X. And then the result of this integral is equals two minus 6/5, which is clearly different from zero, and therefore he is not orthogonal to Q.

So we have some pollen, no meals, and we're supposed to determine a mapping of inner products that's going to define the inner product. And so what I'd like to dio is just use the standard mapping that we learned about in the textbook. Um, so that would be the integral of P one of x times p two of x d x. However, we want this to be a number in the end, and so we must make this a definite interest. Integral eso We could pick really any numbers. Since these air both linear equations, we could pick any numbers. So I'm just going to pick the numbers. Um, negative three and positive three. And so there we have an inner product on first degree polynomial with real coefficients space.


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