Determining whether a set of vectors is a subspace of R^3?

[ParoxysmX]

Homework Statement

Determine whether the set of all vectors of the form (sin2t,sintcost,3sin2t) is a subspace of R^3 and if so, find a basis for it.

Homework Equations

I guess you just need to use the axioms where it is closed under scalar addition and multiplication.

The Attempt at a Solution

If I have two vectors u=(1,2,3) and v=(4,5,6) then u+v = (5,7,9). This gives us 5=sin2t, 7=sintcost, and 9=3sin2t. Am I right in saying there's no (real) value of t which will satisfy any of these equations, meaning (sin2t,sintcost,3sin2t) isn't closed under addition and thus not a subspace of R^3?

 

[HallsofIvy]

Let v be such a "vector". Since a vector space must be closed under scalar multiplication, look at 1000v. Is that still in the set.

(Hint: $-1\le sin(x)\le 1$ for all x.)

 

[mfb]

If I have two vectors u=(1,2,3) and v=(4,5,6) then u+v = (5,7,9). This gives us 5=sin2t, 7=sintcost, and 9=3sin2t. Am I right in saying there's no (real) value of t which will satisfy any of these equations, meaning (sin2t,sintcost,3sin2t) isn't closed under addition and thus not a subspace of R^3?

This would require that u=(1,2,3) and v=(4,5,6) are elements of your set (they are not). You can use the approach posted by HallsofIvy, you just need some vector in the set to begin.

 

[ParoxysmX]

Let v be such a "vector". Since a vector space must be closed under scalar multiplication, look at 1000v. Is that still in the set.

(Hint: $-1\le sin(x)\le 1$ for all x.)

Multiplying by 1000 would give

$-1\le sin(1000x)\le 1$

right? And that statement remains true for all x?

 

[mfb]

While that statement remains true for all x, this is not the point where a multiplication is useful.

Let v be such a "vector".

Start here, please. Can you find such a vector of your set?

 

[ParoxysmX]

Not one with real components I don't think.

 

[mfb]

You have vectors with real components. Actually, all vectors have real components, as you are in R^3. Just plug in some arbitrary value of t - I expect that t is real, so the vector components are real as well.

 

[ParoxysmX]

t=1 say? This would give (0.91,0.45,2.72). Any number for t would give some values back in the form we want.

 

[mfb]

Right. Now you can follow the advice of HallsofIvy and multiply this vector by 1000. Is the resulting vector (how does it look like) part of your set, too?

 

[HallsofIvy]

Multiplying by 1000 would give

$-1\le sin(1000x)\le 1$

right? And that statement remains true for all x?

No! 1000 sin(x) is NOT equal to sin(1000x).

 

[ParoxysmX]

Right. Now you can follow the advice of HallsofIvy and multiply this vector by 1000. Is the resulting vector (how does it look like) part of your set, too?

So that would mean you have 1000sin(x), which no longer oscillates between the same values as sin(x).

 

[mfb]

There is no x here. You get (910,450,2720). Can this vector be part of your set? In other words, is there a t such that (sin2t,sintcost,3sin2t) = (910,450,2720)?
If you can disprove this, you are done.