Which of the following are spanning sets for R3?

[Mathcatchup]

Hi Everyone

I am struggling to understand what is going on here.
1. What does the T exponent stand for?
2. "In part(a), it is easily seen that (a, b, c)T can be written as" Why is that? I don't easily see it.
3. Where did the zero come from? 0(1, 2, 3)T

 

[mmm4444bot]

The forum guidelines explain why full-size screen shots are (in general) unreadable in a post.

We don't really need to see the large gray margins to the left and right, the table of contents, the menu bars, or the icons on your taskbar.

Please display your image on your computer screen first, sized so that the math is easily readable, and then use the Windows Snipping Tool (or your local app) to crop out and save just the relevant part, as a JPG file. (If required, create two smaller images instead of one big image.) Then edit your post, to replace the illegible image with the new image(s).

The forum guidelines also suggest that you proofread what posts look like before submitting them. Thank you! :cool:

 

[stapel]

View attachment 9863

Sorry; I can't read this, either.

1. What does the T exponent stand for?

What definition did the textbook and instructor give, when this notation first arose in class? The exercise is probably using that same definition.

2. "In part(a), it is easily seen that (a, b, c)T can be written as" Why is that? I don't easily see it.
3. Where did the zero come from? 0(1, 2, 3)T

We won't be able to respond to these questions until we can see the exercise.

 

[Dr.Peterson]

I am struggling to understand what is going on here.
1. What does the T exponent stand for?
2. "In part(a), it is easily seen that (a, b, c)T can be written as" Why is that? I don't easily see it.
3. Where did the zero come from? 0(1, 2, 3)T

I would guess that (1,2,3)^T means transpose -- the column vector containing 1, 2, 3. This is mostly a way to save space on the page, especially in problem statements. You can see that in the explanation of part (b), they rewrite the transposes as column vectors.

Once you give us a readable image, or you type out what it says, we can be more sure, but I think (a) is something like this:

r₁, r₂, r₃, (1,2,3)^T
​

where I am guessing that r₁, r₂, r₃ are probably the standard basis vectors (1,0,0)^T, (0,1,0)^T, (0,0,1)^T. Can you see that a(1,0,0)^T + b(0,1,0)^T + c(0,0,1)^T = (a,0,0)^T + (0,b,0)^T + (0,0,c)^T =(a,b,c)^T? Then adding the 0 vector doesn't change it; so you can add 0(1, 2, 3)^T. The idea here is that you can add any additional vector to a spanning set, and the resulting larger set still spans.

A good technique in reading a textbook is to read backward, i.e. use the explanation to see what they meant, by matching up the notation used there (column vectors) with the notation used earlier (transposes). Also, there may be an index of notations in your textbook, which would be a place to look up what T means; otherwise, scan through the definitions in the chapter until you find it. But now, you can just look up the word "transpose".