linear algebra – Vectors which do not belong to column space also pass the test which is the passed by the vectors in column space

In the book Linear algebra and it’s application by Gilbert Strang, it is given that

We have an augmented matrix for system of linear equations Ax=b as:
After applying gaussian elimination:

Description of Column space of A: The column space contains all vectors with b3+b2−5b1=0.
That makes Ax = b solvable, so b is in the column space. All columns of A pass this
test b3 +b2 −5b1 = 0. This is the equation for the plane (in the first description of
the column space).

My question is this: is correct to say that “The column space of A contains all vectors with b3+b2−5b1 = 0”?
I know that all the vectors in column space of A will pass this test b3+b2-5b1 but wont there be vectors outside the column space of A which also pass this test? So, should we say ” the column space of A contains all vectors which pass the test“?

seo – Limit Google site links to just my subdomain when currently they include other sites that don’t belong to me from the same domain

I’ve been struggling to find some information about sitelinks that Google generates when I search for the webapp I’m working on.
enter image description here

As you can see in the image, I’m working on this website, which is a subdomain of The problem is that it generates these sitelinks that have nothing to do with the page.

It’s a simple Angular application and I’ve added all required meta tags and information for SEO, but none of them seems to be related with sitelinks.

Is there a way to do this? A few links i’ve found says that this behaviour is automated, so it means we can’t have any control about it?

event sourcing – In CQRS/ES where does an Aggregate Root belong?

Disclaimer: This question may be related to the framework I’m using to support CQRS/ES rather than the concepts themselves but many of these frameworks implement the same strategies, making me think the two are tightly coupled regardless.

CQRS tells us to…

use a different model to update information than the model you use to read information1

And in event sourcing…

The fundamental idea of Event Sourcing is that of ensuring every change to the state of an application is captured in an event object2

My design includes aggregate root objects, upon which methods are called to make changes (in my particular case called from Commands/Handlers). Those methods check the invariants and then publish an event to a bus, which in turn updates some aspect of the aggregate, typically setting properties or adding items to a collection. These events also update my read model so that I have a projection of the most recent state of the system that can be easily queried. Most of my queries simply act upon the most recent state, but occasionally I need to create a projection for an aggregate as it existed at a point in the past (hence the use of event sourcing).

As such my aggregate root and read model share a very similar “shape”, so similar that I’ve created an interface that both implement so that I can treat them equally depending on the type of query being executed.

Given that the aggregate root and read model are so similar, and even though the aggregate appears to belong to the write model (as the commands act upon it) does it in fact belong in the read model?

Or, where does the aggregate root belong? In the write model, the read model or in a shared domain model, which seems to go against the whole CRQS idea?

Polynomial time optimization problems belong to which complexity class?

Take a look at the FP complexity class.

Technically speaking, it is wrong. However, usually people don’t make a huge difference between the two. Its usually clear from the context whether the problem is a decision or a search problem, hence its clear whether it should be in $P$ or $FP$.

So, writing that “Finding the shortest $(s,t)$ path in the graph $G$, is a $P$ problem” would actually mean: “Finding the shortest $(s,t)$ path in the graph $G$, is an $FP$ problem” (since the context is clear)

sql server – How to model a course > section > lesson structure where lessons can belong to either sections or courses?

I have three entities:

  1. Course
  2. Section
  3. Lesson

Each “course” is made up of several “lessons”.
The “lessons” inside a course, can either be categorized into “sections”, or not.

So, the contents of a course could look either like this:

Foo Course:
    Lesson 1
    Lesson 2
    Lesson 3
    Lesson 4

Or like this:

Bar Course:
    Section 1:
        Lesson 1
        Lesson 2
    Section 2:
        Lesson 3
        Lesson 4

So, in other words, a course can either directly have “lessons”, or it can have “sections” that in turn have “lessons”.

From the other perspective, a “lessons” can either directly belong to a “course”, or belong to a “section” that in turn belongs to a “course”.

I’m struggling with how to implement this structure in a a relational database.

If every “lesson” had to necessarily belong to a “section”, it would be easy, I could just simply have a “Course” table, a “Section” table with a “CourseId” column, and a “Lesson” table with a “SectionId” column.

But my scenario is not as straightforward as that. A “section” can potentially exist as a middleman between a “course” and several “lessons”, but it can also be absent, in which case a “course” directly has the lessons and no there are no “sections”.

I’d appreciate any suggestions regarding how such a structure can ideally be implemented in the context of relational databases.


tokens – How can I watermark images with a taxonomy term from the node they belong to?

I’m working on a site for a business who would like to watermark all their product photos with the brand name of the product (which is a taxonomy term reference field on the product content type).

Looking into this, I found a variety of modules that will watermark images in various ways, but most expect to just put a single image or text overlay on all images of a given style, which is not what I need here.

The one exception I found was Textimage, which works with the Image Effects module and, at least in theory, supports node tokens. But when I tried out those two modules, I was unable to get any token for the taxonomy field in question to work. I tried (node:field_brand) and (node:field_brand:entity:name) without success — it just put the text of the token instead of replacing it with the brand. Other tokens, like (node:title), did work, but not the one I wanted (even though it was in the list of available tokens).

I posted to the issues queue for Textimage, but it doesn’t seem like anyone’s monitoring it regularly, and I’m on a deadline.

Does anyone know either how I can get the node replacement to work in Textimage, or alternatively any other way of doing this? I’m not at all wedded to Textimage and Image Effects — they’re just the only modules thus far that claim to be able to do what I need to do.

machine learning – What computational complexity classification does autonomous/self-driving cars belong to?

Self-driving car technology continues to attract popular attention and interest in today’s media, but how would a computer scientist explain the theoretical nature of the problem? For example, we are frequently told that Sudoku and Go are examples of NP-completeness and PSPACE-completeness, and a range of practical progress has been made in dealing with these two categories. So similarly with autonomous vehicles, I think laypeople people (like myself) intuitively accept that learning to drive properly is a “problem”, but then, what is the underlying problem that such machine learning algorithms are trying to tackle?

I did some Googling around and found different articles and papers alluding to some parts or formal abstractions of the driving problem and/or maybe robotics being PSPACE-complete. So is that the answer basically? Or is it that the question is too vague to be answerable? Or even that there hasn’t been a clear understanding yet, because the area is fairly new? I’m just really curious, but didn’t find anything explaining this at a high level. But when I took CS in college, the professors would emphasize to programmers that understanding the nature of the problem can be very important, e.g. understanding if your specific problem is in P or is NP-complete, and so forth.

complexity theory – Must an optimization problem with a greedy algorithm belong to P?

If it is known that for some optimization problem there is a greedy algorithm that solves it and the solution includes sorting of input at the preliminary stage, is it necessarily true that the problem belongs to P?

I was told it is true that problems of this type must belong to P, but I have questions about whether this is true.

It seems to me that there should be many examples of optimization problems with non-deterministic greedy algorithms. Possibly for TSP?

Are there in fact counterexamples to this claim or is it necessarily true?

real time – Do 1:1 mobile video call face filters belong on Client or Server?

I am drafting architecture for 1:1 video calling mobile app with face filters (face recognition).

I could use p2p e.g. WebRTC because more savings less complexity. However, continuous face rec during a video call on cheap phones becomes more expensive for the device.

Is this cost increase generally significant enough to warrant routing the traffic through a server?

copay – Does private key belong to one publick key in Bitcoin?

Each private key corresponds to one public key, but a mnemonic key contains the master private key which describes how each private key is generated.

If you want to withdraw from it, I would use Electrum. It can show you the addresses and you can export individual private keys.