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I am currently studying an online linear algebra book.

The following definition for the vector form of a line is given:

Vector Form of a Line. Let $\ell$ be a line and let $\vec d$ and $\vec p$ be vectors. If $\ell=\{\vec x : \vec x= t\vec d+\vec p\text{ for some } t\in \mathbb{R} \}$, we say the vector equation $$\vec x=t\vec d+\vec p$$ is $\ell$ expressed in vector form. The vector $\vec d$ is called a direction vector for $\ell$.

The following section comes shortly after (the colors are my addition, so I can easily refer to the highlighted portions later):

It's important to note that when we write a line in vector form, it is a specific shorthand notation. If we augment the notation, we no longer have written a line in "vector form".

Example. Let $\ell$ be a line, let $\vec d$ be a direction vector for $\ell$, and let $\vec p\in \ell$ be a point on $\ell$. Writing $$\color{blue}{\vec x=t\vec d+\vec p}$$ or $$\color{blue}{\vec x=t\vec d+\vec p\quad\text{ where }\quad t\in \mathbb{R}}$$ specifies $\ell$ in vector form; both are shorthands for $\{\vec x : \vec x=t\vec d+\vec p\text{ for some }t\in \mathbb{R} \}$. But, $$\color{red}{\vec x=t\vec d+\vec p\quad\text{ for some }\quad t\in \mathbb{R}}$$ and $$\color{red}{\vec x=t\vec d+\vec p\quad\text{ for all }\quad t\in \mathbb{R}}$$ are logical statements about the vectors $\vec x$, $\vec d$, and $\vec p$. These statements are either true or false; they do not specify $\ell$ in vector form.

Similarly, the statement $$\ell = t\vec d+\vec p$$ is mathematically nonsensical and does not specify $\ell$ in vector form. (On the left is a set and on the right is a vector!)

I don't understand why the red lines are unacceptable as specifying $\ell$ in vector form.

They seem to be equivalent to the second blue line, which is a bit ambiguous and could stand for either of the red lines.

I also don't understand why the red lines being true or false statements disqualifies them as a shorthand for a set. As far as I can tell, the blue line equalities are also logical statements (stating that the left-hand-side of an equation is equal to the right-hand-side is a true or false statement). And there's nothing wrong with any of these logical statements, because the long form of $\ell$, a set, contains a logical statement within the set's definition, and we are stating that logical statement from the set's definition as a short hand.

What am I missing here?

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    $\begingroup$ If I ask you for the line between points $A$ and $B$. "True" or "false" is not an answer to that question. $\endgroup$ – Derek Elkins left SE Sep 18 '19 at 21:18
  • $\begingroup$ @DerekElkins I agree, but my point is that all four highlighted statements are "true or false" statements, representing the "true or false" statement from within the long-form definition of a line. Why is it okay to use some, and not others? $\endgroup$ – Calculemus Sep 18 '19 at 21:29
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    $\begingroup$ That said, I don't really agree that the blue text is shorthand for the set comprehension. If anything, the blue text is most commonly (ab)used for defining a function $\vec x(t)$ which is related to but distinct from the set (which would be the range of this function). Nevertheless, if the first blockquote is a definition, then the example would be using the definition. $\endgroup$ – Derek Elkins left SE Sep 18 '19 at 21:30
  • $\begingroup$ @DerekElkins The first blue line is defined, in the given definition, as the short-hand for the vector form of a line. I think that if we follow that definition, regardless of what the general convention may be, then the first statement is acceptable as a shorthand. But even so, I still don't understand why the second blue statement is acceptable, and the red statements aren't. $\endgroup$ – Calculemus Sep 18 '19 at 21:34
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    $\begingroup$ I think what's going on here is that the blue formulas are already somewhat incomplete and technically improper, and they are recognized as "lines" only by virtue of people (or at least, the author of this book) accepting these exact forms in place of a really correct description of a line. You might think of them as mathematical idioms that do not literally mean what they say and that cannot be rephrased in ways that less idiomatic expressions can be. $\endgroup$ – David K Sep 19 '19 at 3:45
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To start, none of the colored formulas would represent the set $\{\vec x\mid \exists t\in\mathbb R.t\vec d + \vec p = \vec x\}$ in a general context. The closest, if anything, would be the first red formula which would represent the predicate $P(\vec x)\equiv\exists t\in\mathbb R.t\vec d + \vec p = \vec x$ which gives rise to the previously mentioned set via comprehension. Probably the key thing to note here is that a set is an object that we manipulate, like a number, not a statement whose truth we attempt to establish.

The passage as a whole is pretty confusing since it doesn't seem to explain what's really going on in the blue statements.

What seems to be going on here is that it is very common to write something like $y=ax+b$, say, with the intent that we're actually defining a function called $y$, so that it would be more accurate to write $y(x)=ax+b$. Given this reading of the blue statements, then what the authors seem to intend is that a line is the range of this function. Explicitly, the line that the function would induce is $\{\vec x(t)\mid t\in \mathbb R\}$. That said, it is important to distinguish the line from the function (which this notation does not do). $\vec x(t)$ and $\vec x(g(t))$ where $g:\mathbb R \to \mathbb R$ is some bijection are distinct functions (unless $g$ is the identify function) but give rise to the same line. As concrete example, $\vec x(t)$ and $\vec x(2t)$ are different functions but the latter just corresponds to moving along the same line twice as fast, to use a kinematic metaphor.

With this reading, the blue formulas define a function $\vec x$, and this function is an object, not a statement. This object is closely related to (but distinct from) the set that lines are defined to be.

The red formulas can't be interpreted as functions in this way.

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  • $\begingroup$ $\vec x$ would generally be called a parameterization of the line, but we usually want the line to be independent of how it's parameterized. $\endgroup$ – Derek Elkins left SE Sep 18 '19 at 22:01

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