Is the span of the infinite set S the same as the span of a finite set?

[baddin]

Homework Statement

Give S = {(x,|x|,2|x|) | x \in R} \bigcup {(0,2,4),(-1,3,6)}, find span(S)

Homework Equations

I know that span of a finite set of vectors is given by <a(0,2,4) + b(-1,3,6)+c(x,|x|,2|x|)>, where a,b,c are any real numbers. Can i use that same way to find the span of this infinite set.

The Attempt at a Solution

Is the solution same as the vector span for a finite set like span(S) = <a(0,2,4) + b(-1,3,6)+c(x,|x|,2|x|)>, or is it something else?

 

[LCKurtz]

Homework Statement

Give S = {(x,|x|,2|x|) | x \in R} \bigcup {(0,2,4),(-1,3,6)}, find span(S)

Homework Equations

I know that span of a finite set of vectors is given by <a(0,2,4) + b(-1,3,6)+c(x,|x|,2|x|)>, where a,b,c are any real numbers. Can i use that same way to find the span of this infinite set.

The Attempt at a Solution

Is the solution same as the vector span for a finite set like span(S) = <a(0,2,4) + b(-1,3,6)+c(x,|x|,2|x|)>, or is it something else?

Lots of those x vectors are multiples of each other. I would start by looking at what S looks like for x > 0 and x < 0.

 

[baddin]

So for any values of x I pick. the x vectors will be linearly dependent and they cannot form my span? So would that mean the span(S) = span of linearly independent independent vectors in S. So span(S) = <a(0,2,4)+b(-1,3,6)>?

 

[LCKurtz]

So for any values of x I pick. the x vectors will be linearly dependent and they cannot form my span?

I don't know what you are trying to say here. I will say it again: what do the x vectors look like if $x > 0$ versus $x<0$? You might start by actually answering that question.

 

[baddin]

So for x<0 The x vectors look like(x,-x,-2x) and for x>0 the x vectors look like (x,x,2x).

 

[LCKurtz]

And if you factor an x out of each what happens? And why do you say they are linearly dependent?

 

[baddin]

OH my fault, I see they are linearly independent. So could I generalize this and write span(S) = {a(1,1,2) + b(1,-1,2) + c(0,0,0) + d(0,2,4) + e(-1,3,6)} given a>0, b<0, c,d and e are any real numbers?

 

[LCKurtz]

OH my fault, I see they are linearly independent. So could I generalize this and write span(S) = {a(1,1,2) + b(1,-1,2) + c(0,0,0) + d(0,2,4) + e(-1,3,6)} given a>0, b<0, c,d and e are any real numbers?

Is that b vector correct?

Usually when you are asked to describe a span you wouldn't include extra vectors that don't add anything. Since these are 3D vectors you would expect at most to need 3 vectors and maybe fewer to get an independent spanning set. And remember that a span automatically doesn't restrict the multiplying constants to positive or negative.