Does the Set S Span P3? Solving for Linear Span in Vectors

[eyehategod]

Determine whether the set S={x$$^{2}$$-2x,x$$^{3}$$+8,x$$^{3}$$-x$$^{2}$$,x$$^{2}$$-4} spans P3

for example, if i had S={(123),(234),(345)} and asked to determin if this set spans R$$^{3}$$, then I would write these components as a matrix and then reduce to echelon form. From there Id be able to tell if this set spans R$$^{3}$$.

But for this problem I don't know how to approach it. Its the first time i see a problem like this.

 

[cristo]

Well, first you need to think what P3 is (I don't know what you mean by it!), and then you need to recall your definition of a spanning set. Then you need to have a go at proving it, and post what you get!

 

[JasonRox]

Do as cristo says. Come back with as much information as you can get. If you still don't get it, I'll offer some more help and a way (probably better way) to thinking about spanning sets.

 

[eyehategod]

goddammit I am starting to hate this forum.

 

[cristo]

goddammit I am starting to hate this forum.

Well, that's a good way to get others to give up their free time to help you . We're not here to give out answers to homework questions.

 

[eyehategod]

i tried the method that i posted first and I am getting that this set does span P3. Is this the right answer?

 

[JasonRox]

goddammit I am starting to hate this forum.

We have a choice to either A... feed you the solution so you can pass the assignment, or B... help you to solve problems so you can pass the class. B is the wise choice. And yes, wise actions and answers are a ***** sometimes.

 

[JasonRox]

i tried the method that i posted first and I am getting that this set does span P3. Is this the right answer?

If it doesn't span R^3, then you must also show this too. You can't just say... no it doesn't span R^3 and show no work.

All we're asking is that you check what it means to span R^3. If you don't really know what spanning is, you'll never able to show any work or solve any problems relating to spanning a set.

I was a TA for Linear Algebra and no one knew anything about spanning sets and basically all lost marks or didn't get any at all. For those who showed up to my seminar and listened to my straight up way of working with spanning sets, they did well.

I have a question and see if you can answer it. It requires no work at all.

Do the following vectors (1,2) and (2,3) span a set?

 

[eyehategod]

it spans R^2. the only method i know requires me to write these vectors in matrix form.
i would ge:
1 2
2 3

reduced to echelon form i would get

1 2
0 -1

this tells me the set Spans R^2.
if a row had been all 0s, then the set would span R^1.
is this approach correct?

 

[JasonRox]

it spans R^2. the only method i know requires me to write these vectors in matrix form.
i would ge:
1 2
2 3

reduced to echelon form i would get

1 2
0 -1

this tells me the set Spans R^2.
if a row had been all 0s, then the set would span R^1.
is this approach correct?

Yes, it would be correct to say that the spanning set is isomorphic to R^1, but not that is spans R^1.

Now, check if the set you have now spans P_3, or even R^3 since it's basically the same because... 3x^2 + 1 can be written in the vector form (3,0,1), right?

 

[eyehategod]

if any vector in P_3 can be written as a linear combo of the vectors in S, then can i conclude that the set S spans P_3?

 

[JasonRox]

if any vector in P_3 can be written as a linear combo of the vectors in S, then can i conclude that the set S spans P_3?

Exactly!

 

[eyehategod]

eureka!

 

[HallsofIvy]

if any vector in P_3 can be written as a linear combo of the vectors in S, then can i conclude that the set S spans P_3?

That is, in fact, the definition of "span". It's a good idea to spend more time learning definitions than learning methods of solving specific problems.

 

[Chris Hillman]

Homework Help

BTW, eyehategod, this looks like a homework question. If so, next time you should post HW questions in the "Homework Help" forum at PF, which has some special rules.