2. Vector allocation: in this step, we allocate memory for the vector. For most vectors, wehave two options: 1) if we know the maximum vector capacity, we can specify it by calling theallocateNew(int) method; 2) otherwise, we should call the allocateNew() method, and a defaultcapacity will be allocated for it. For our running example, we assume that the vector capacity neverexceeds 10:

3. Vector mutation: now we can populate the vector with values we desire. For all vectors, we can populatevector values through vector writers (An example will be given in the next section). For primitive types,we can also mutate the vector by the set methods. There are two classes of set methods: 1) if we canbe sure the vector has enough capacity, we can call the set(index, value) method. 2) if we are not sureabout the vector capacity, we should call the setSafe(index, value) method, which will automaticallytake care of vector reallocation, if the capacity is not sufficient. For our running example, we know thevector has enough capacity, so we can call

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5. Vector access: it is time to access vector values. Similarly, we have two options to access values:1) get methods and 2) vector reader. Vector reader works for all types of vectors, while get methods areonly available for primitive vectors. A concrete example for vector reader will be given in the next section.Below is an example of vector access by get method:

The steps are not necessarily performed in a linear sequence. Instead, they can be in a loop. For example,when a vector enters the access step, we can also go back to the vector mutation step, and then set valuecount, access vector, and so on.

We should try to make sure the above steps are carried out in order. Otherwise, the vectormay be in an undefined state, and some unexpected behavior may occur. However, this restrictionis not strict. That means it is possible that we violates the order above, but still getcorrect results.

When mutating vector values through set methods, we should prefer set(index, value) methods tosetSafe(index, value) methods whenever possible, to avoid unnecessary performance overhead of handlingvector capacity.

All vectors implement the AutoCloseable interface. So they must be closed explicitly when they areno longer used, to avoid resource leak. To make sure of this, it is recommended to place vector related operationsinto a try-with-resources block.

For fixed width vectors (e.g. IntVector), we can set values at different indices in arbitrary orders.For variable width vectors (e.g. VarCharVector), however, we must set values in non-decreasing order of theindices. Otherwise, the values after the set position will become invalid. For example, suppose we use thefollowing statements to populate a variable width vector:

A ListVector is a vector that holds a list of values for each index. Working with one you need to handle the same steps as mentioned above (create > allocate > mutate > set value count > access > clear), but the details of how you accomplish this are slightly different since you need to both create the vector and set the list of values for each index.

This result will be a new vector (for example, an IntVector) that can act in place of the original vector (for example, a VarCharVector). When you write the data in arrow format, it is both the new IntVector plus the dictionary that is written: you will need the dictionary later to retrieve the original values.

As far I have understood the math in the question, the vector is p, the remaining part is just an index to the vector. Therefore, a short vector would be sufficient as shown in the following example. I do not know about e, s2, and d7, but 10 Hz is clearly a number with unit, typeset upright, as already shown in the question and correctly done via \SI{10}{\hertz} with the macro \SI from package siunitx.

You might want to give a try to the esvect package, which has 8 nice arrow shapes, longer than the arrow that comes with vec. The base command is \vv, and there is a \vv* command for vectors with indices, in order to have a correct spacing between vector and index:

I ask this because in my linear algebra class, my professor never used arrow notation, so sometimes it wasn't obvious between distinguishing a scalar and a vector. (Granted, he did reserve $u$, $v$, and $w$ to mean vectors.) At the same time, my machine learning class used arrows to denote vectors, but I know some other machine learning literature chooses not to put arrows on top of their vectors.

Let us decide we mark an element $\mathbb{R}^3$ as a vector, so we write $\vec{v}$ for it.Now, we want to multiply it with a $3\times 3$ matrix, since it is a matrix there is no arrow, or is there? After all the $3\times 3$ matrices form a vector-space and sometimes we use that structure. So, $\vec{A}$?

For example when we show that for $P$ the characteristic polynomial we have $P(\vec{A})= 0$. Wait, haven't we seen the polynomials as an example of an infinite dimensional vector space? Should be put an arrow there, too? Then we can have $\vec{P}(\vec{A})\vec{v}$!

I tried to write this a bit playful. But the serious point is that one really switches the point of view somewhat frequently when doing mathematics, and the notion of 'vector' is not so clear cut, as plenty of structures are (also) vector space.

In somewhat advanced (pure) mathematics, it is thus not very common to use the notation with an arrow to mark elements as vector specifically. But, if in some context it seems useful, there is no problem with it either.

Usually, it's fine to not have an arrow over your vectors as long as you define that they are vectors. Although in any case, really, you should define it to be a vector with or without an arrow. Once you say "Let v be a vector" then no arrow is needed. If I remember correctly, one of my linear algebra professors didn't use arrows on theirs while my other professor who is an algebraist uses arrows. If you're using a lot of scalars and vectors, using arrows might be handy. Again, it's a matter of preference, convenience, and the "situation" you're in. If there were numerous scalars and vectors which I was dealing with, I would use arrows so it's easier to spot which is a vector and which is not.

Notation indicates some mathematical maturity but it doesn't say much. I think precision is a greater factor. A "mature" mathematician might put an arrow over v without defining it (though who are we kidding, I doubt such a mathematician exists -- it is mediocre practice). A more mature mathematician would define what they mean by v-arrow (or simply v) at the get-go. So define what you mean and you will be safe.

A $2\times2$ real matrix is a vector in the vector space of $2\times2$ matrices. The polynomial $t^2+t+3$ is a vector in the vector space of polynomials of degree at most three. The function $\sin(x)$ is a vector in the space of continuous functions. Should we put arrows on these?

I think the confusion lies in the term "vector". Most people think first of an element in $\mathbb{R}^n$ when they hear vector - maybe they were introduced to vectors like that in school, or they are just the most common they work with. And then the arrow above the symbol makes perfect sense.

But in mathematics, a vector is defined as an element in a vector space, and that can be pretty much any structure. $\mathbb{R}^n$ is just the most common example, but functions, matrices, every field (so in particular scalars), and many more are vectors as well.

I think it's a maturity issue in one sense, which is that when you're first introduced to vectors is likely to be the first time you're exposed to the fact that there are two different operations, both called multiplication and both written the same way. You need to learn that you can multiply scalar by scalar, or vector by scalar, but not vector by vector. And perferably understand why. Same issue for addition, you can add two scalars or two vectors, but not a vector to a scalar.

However, once you understand the difference and have become accustomed to distinguishing them in your mind, it's less useful to write them very differently, and might become an annoying overhead. Reserving letters still helps, and usually is enough to remember what everything is. So, putting arrows or underlines everywhere feels like too much. In typeset text you might continue to use bold for vectors because it doesn't make things look much more busy on the page, but in writing I think most people drop it.

Furthermore, like quid's answer says, you will eventually need to deal with more than one vector space at once, at which point having a notational convention that distinguishes "vectors in my vector space" from "everything else" is not sufficient anyway. So as you mature you have to rely on other means to understand what type of entity everything is.

Perhaps off-topic, but there's a related thing in computer programming called "Systems Hungarian notation", where every variable name contains a prefix denoting the data type of the variable. Almost everyone hates this, because in practice it's too much repetition of the same information. It's actually quite difficult to strike the "right" balance because it's a matter of taste. Using mathematical block capitals for special entities $\mathbb{R}$, latin vs. greek letters for vectors vs. scalars, $n$ for an integer, $q$ for a rational, $x$ for a real, $v$ for a vector, $a$ for a coeffient and $f$ for a function, uppercase for matrices, all seem perfectly sensible to most mathematicians although none is essential. Hanging decorations off your letters, though, starts to cross a line and so fewer people bother.

I will buck the consensus here and say that yes, all else being equal, putting arrows on vector quantities is a slight signal that either the author or the intended audience are beginning students or physicists.

In my experience, $\overset{\rightharpoonup}{v}$ and $\mathbf{v}$ are used more often in science and engineering (even at the research level). This explains why they are often used in lower division mathematics courses, since at this point most of the class is not a mathematics major. As you proceed through mathematics, you begin to see vector spaces like real numbers and function spaces, and it makes less sense to think of vectors as "different" from the other things you're working with. Once you start hearing the word "module" enough, the notation has almost completely disappeared. The only exceptions I have seen are in differential geometry and applied math. I don't think the notation is so much mathematically immature as it is associated with "spatial" thinking. 0852c4b9a8

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