A Linear Transformation Problem

[The Student]

It's probably just a very simple problem, but I don't understand the first sentence.

[Problem: The line segment from 0 to a vector u is the set of points from tu, where 0 ≤ t ≤ 1. Show that a linear transformation T maps this segment into the segment between 0 and T(u).]

Is u a vector lying on 0, or is it just arbitrarily somewhere that doesn't go through 0?

It says that the points go "from tu"; okay, but from tu to what? Maybe it meant "from 0 to tu"?

 

[HallsofIvy]

I think you are misunderstanding the problem. We can think of a vector as a set of points extending from 0, the origin, to u. If T is any linear transformation then, of course, T applied to the point 0 gives T(0)= 0 and T applied to the point u gives T(u). You are to show that T applied to any point between 0 and u gives a point between 0 and T(u). To put it another way, linear transformations map line segments to line segments.

 

[The Student]

I think you are misunderstanding the problem. We can think of a vector as a set of points extending from 0, the origin, to u. If T is any linear transformation then, of course, T applied to the point 0 gives T(0)= 0 and T applied to the point u. You are to show that T applied to any point between 0 and u gives a point between 0 and T(u). To put it another way, linear transformations map line segments to line segments.

Are you saying that u is a point? It said that u is a vector; is that what they call a position vector? So is it geographically correct to say that u is a line from the origin to the point u?

 

[pka]

Are you saying that u is a point? It said that u is a vector; is that what they call a position vector? So is it geographically correct to say that u is a line from the origin to the point u?

As Prof HofI said, you are reading too much into the word vector.
A vector is really a hybrid. A vector is a concept that involves both length and direction.

It is usual to use the notation \(\displaystyle \vec{u}=<a,b,c>\) for a vector.
We also use \(\displaystyle u=(a,b,c)\) as the notation for the point u.
But think about; we can say a point has length, its distance from the origin.
It has a direction from the origin.
So it is reasonable to identify \(\displaystyle \vec{u}\) as the position vector of the point u

 

[The Student]

As Prof HofI said, you are reading too much into the word vector.
A vector is really a hybrid. A vector is a concept that involves both length and direction.

It is usual to use the notation \(\displaystyle \vec{u}=<a,b,c>\) for a vector.
We also use \(\displaystyle u=(a,b,c)\) as the notation for the point u.
But think about; we can say a point has length, its distance from the origin.
It has a direction from the origin.
So it is reasonable to identify \(\displaystyle \vec{u}\) as the position vector of the point u

Okay, now does the phrase "from tu" make sense in the first sentence, "The line segment from 0 to a vector u is the set of points from tu, where 0 ≤ t ≤ 1"? Should it have been just "...is the set of points in tu"? That is what is confusing me the most. Maybe "in" is not good either, but "from" doesn't seem to make any sense at all.

 

[pka]

Okay, now does the phrase "from tu" make sense in the first sentence, "The line segment from 0 to a vector u is the set of points from tu, where 0 ≤ t ≤ 1"? Should it have been just "...is the set of points in tu"? That is what is confusing me the most. Maybe "in" is not good either, but "from" doesn't seem to make any sense at all.

Yes of course.
Many authors make no distinction between the point u and the vector \(\displaystyle \vec{u}=<a,b,c>\). I am one that does.

In scalar multiplication this is the idea \(\displaystyle t\vec{u}=<ta,tb,tc>\) where \(\displaystyle t\in\mathbb{R}\) (or complex).

Now if \(\displaystyle t=0\) then \(\displaystyle t\vec{u}=<0.0,0>\) ; if \(\displaystyle t=1\) then \(\displaystyle t\vec{u}=<a,b,c>\).
That is the line segment from the origin to the point u =(a,b,c). So there you have it. O.K.?

 

[The Student]

Yes of course.
Many authors make no distinction between the point u and the vector \(\displaystyle \vec{u}=<a,b,c>\). I am one that does.

In scalar multiplication this is the idea \(\displaystyle t\vec{u}=<ta,tb,tc>\) where \(\displaystyle t\in\mathbb{R}\) (or complex).

Now if \(\displaystyle t=0\) then \(\displaystyle t\vec{u}=<0.0,0>\) ; if \(\displaystyle t=1\) then \(\displaystyle t\vec{u}=<a,b,c>\).
That is the line segment from the origin to the point u =(a,b,c). So there you have it. O.K.?

It's just that the phrase "from tu" still does not make sense; "from tu" to what? For example, if u = <1, 3> then tu = <t1, t3>. So from what I understand, we would basically have a line that would look like y = x/3 on an x-y plane but stops at one unit up and three units to the right because 0 ≤ t ≤ 1. So if I have all of that right, why wouldn't it say that the set of points are "in tu" instead of "from tu"?

 

[HallsofIvy]

The original problem, as you wrote it, said "\(\displaystyle 0\le t\le 1\)". What is "tu" if t= 0? If t= 1?

 

[The Student]

The original problem, as you wrote it, said "\(\displaystyle 0\le t\le 1\)". What is "tu" if t= 0? If t= 1?

So is the problem worded poorly?

Assuming that the wording of the problem is fine, would this geographically represent a line u from the origin to some arbitrary point scaled down by the weight of t?

 

[HallsofIvy]

No, the question is worded correctly. No, it does not "represent a line from the origin to some arbitrary point scaled down by the weight of t". t is a variable that can take on any value between 0 and 1 inclusive. It is a specific line segment. Can you not answer the questions in my previous post?

 

[The Student]

No, the question is worded correctly. No, it does not "represent a line from the origin to some arbitrary point scaled down by the weight of t". t is a variable that can take on any value between 0 and 1 inclusive. It is a specific line segment. Can you not answer the questions in my previous post?

Thank-you so much for your patience! : )

What is "tu" if t= 0? If t= 1?

Here are my answers to your questions. For t = 0: 0*u = 0. And for t = 1: 1*u = u.