Doubt About Inner Product

The answer to your question:is that we can define any basis to be an orthonormal basis by choosing the right inner product?is Yes. two vectors are defined orthogonal if its inner product is null, so the notion of orthogonality is ''inner product dependent''. And any basis is orthogonal with respect a suitable inner product. About the same problem you can see the answers to my question: What really is ''orthogonality''?

Doubt About Inner Product 1

1. Why is this inner product sensible?

I've got the book (Finite Dimensional Vector Spaces).He's talking about why the inner product is a better representaion of the properties of two vectors than the angle between them. In the preceeding paragraphs he describes an inner product in $R^2$, where the angle can range from 0 to $2pi$. When you look at $R^1$ then the angle can only be 0 or $pi$ while the inner product can be any value. So he says that the inner product shows greater sensitivity.Hope the explanation helps, I am not sure what conclusion I can draw from Halmos's statement. If it gives you any deep insight into the universe please share it with us

2. Is there an "inverted” dot product?

I would write$$ mathbfa star mathbfb = prod_i=1^n left(a_i b_i

ight)=(a_1 b_1),(a_2b_2)cdots ,(a_nb_n) $$up as$$ mathrmprodleft(mathbfamathbfb

ight), $$where $mathrmprodleft(

ight)$ is the array product function,$$ mathrmprodleft(mathbfx

ight)equivprod_foralltextindicesix_i. the nice properties of this operation follow from the simplicity of this construction. There are $p$-norms,$$ left|left|mathbfx

ight|

ight|_pequivleft(sum_forallileft|x_i

ight|^p

ight)^p^-1, $$which is more generally defined as:An array product can be written in this form where the element-wise transform is $lnleft(

ight)$ and its reverse is $expleft(

ight)$, i.e.$$ left|left|mathbfx

ight|

ight|_textlogequivexpleft(sum_forallilnx_i

ight). $$ So, this might be written up as$$ left|left|mathbfamathbfb

ight|

ight|_textlog, $$which might be called the "log-norm" by analogy to $p$-norms.This occurs in product topology, so we might call it the "product norm".From a practical perspective, I would try to work in the log-transformed domain when possible. I mean,$$mathrmprodleft(mathbfamathbfb

ight)$$is a nice, clean expression, but$$mathbfamathbfb$$is really clean.It would be easier to describe exactly how to do this if your problem domain were known, but an abstract gist would be like:For precedents, this sort of technique is widely used in specific cases like Laplace transforms and Fourier transforms. So, it seems like you should basically try to create your own log-scale transform domain appropriate to the problem.

Doubt About Inner Product 2

3. Catalog promotions - Discount appear in product list but not in product view

OK, now I am really scared. I worked on the same issue 3 days ago. What were the odds? This may be the reason: Because the promotions (and special prices) can be set to have a time limit, the product prices need to be rebuilt each day, to check if the promo price is still available. This seams to work for me: I did not have the cron setup for my magento instance. (Rookie mistake that I am ashamed of admitting). Add this to your crontab:

4. the use of Cartesian product

The Cartesian product of $2$ sets $A$ and $B$ is just the set of all ordered pairs $(a,b)$ where $a in A$ and $b in B$. You can think of it as creating a set of from 2 other sets. For example $A = B = mathbbR => Atimes B = mathbbR^2$. Put two real number lines perpendicular to each other and you get the xy-plane. Another example is $A=(6,8), B=(1,2,5) => A times B = ((a,b))$ where a could be either 6 or 8 and b could be 1, 2 or 5. Thus an element of $Atimes B$ could be $(6,2)$, but not $(8,7)$ since $7$ is not in $B$. Some branches of mathematics use "The Axiom of Choice" which is equivalent to "the cartesian product of a collection of non-empty sets is non-empty". Said Axiom is used in measure theory to prove that there exists sets that cannot be "measured"

5. the best creatine product?

I used to take the GNC brand creatine monohydrate its powder form and it works great. When I started working out and taking it really makes you pumped and more energy. Good Luck

6. How to check if the product is in a certain category on a single-product.php in Woocommerce?

I would look at using the get_categories() function of the WC_Product class.You can find the link to the documentation here.Basically within the loop of the page call the function to return the categories associated with the product.

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