Jump to content

# Linear transformations

## Recommended Posts

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A).

Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?
(I can do this with numbers but the trig is throwing me).

##### Share on other sites

I think you have missed out an ex before the second sinx.

Looking then at your transformation, and noting that the domain and codomain of, are the same, (so the transformation is closed),  you can surely test against the linear space axioms by forming the derivatives and sums and showing there are inverses etc.

##### Share on other sites
23 hours ago, Lauren1234 said:

Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?

The assignment is wrong. It is like asking how to prove that $$f: [0,1]\to [0,1],\, x\mapsto 2x$$ defines a linear function $$f$$, when $$[0,1]$$ means the closed interval of real numbers between $$0$$ and $$1.$$

Edited by taeto
##### Share on other sites
1 hour ago, taeto said:

The assignment is wrong. It is like asking how to prove that f:[0,1][0,1],x2x defines a linear function f , when [0,1] means the closed interval of real numbers between 0 and 1.

I think perhaps there have been some transciption errors in the OP?

##### Share on other sites
14 minutes ago, studiot said:

I think perhaps there have been some transciption errors in the OP?

Something is wrong, I may not have pinpointed it exactly. The notation is suspect; $$A$$ is a set of what kind of elements, and then what is span($$A$$)? It makes sense if we read the OP as $$A = \{\exp,\exp \cos, \exp \sin, \cos, \sin \}$$, and taking span($$A$$) to be the space of all linear combinations with real coefficients of those five functions. Then the only thing to prove is that the image of the span under $$T$$ is contained in the span. Hence my original comment.

Edited by taeto
##### Share on other sites

Sorry it should be

Let A={e^xsin(x),e^xcos(x),sin(x),cos(x)}

##### Share on other sites
4 hours ago, Lauren1234 said:

Sorry it should be

Let A={e^xsin(x),e^xcos(x),sin(x),cos(x)}

That makes sense. The meaning is that $$A$$ is a set of four functions. The role of $$x$$ remains a little ambiguous. If we tread lightly, we can surmise that $$x$$ is a coordinate, and its range is the reals.

Do you see what is the span($$A$$)$$=V$$? Is it true that if a function $$f$$ is any one of the four functions in $$A,$$ then $$f'$$ belongs to $$V?$$ And what if $$f$$ is any function in $$V?$$

##### Share on other sites
• 4 months later...

Linearity is one thing. That's inherited from linearity of the derivative operator.

Closure is another thing. That's what Studiot and Taeto are talking about, I think.

Because ex, exsin x, excos(x), sin(x), cos(x) are meromorphic (analytic in C ==> analytic in all R) I see no problem with the domains.

Closure can be assured by inspection. The most involved cases are exsin(x), excos(x). But,

$\frac{d}{dx}e^{x}\sin x=e^{x}\sin x+e^{x}\cos x\in\textrm{span}\left(A\right)$

$\frac{d}{dx}e^{x}\cos x=e^{x}\cos x-e^{x}\sin x\in\textrm{span}\left(A\right)$

That would be my answer.

## Create an account or sign in to comment

You need to be a member in order to leave a comment

## Create an account

Sign up for a new account in our community. It's easy!

Register a new account

## Sign in

Already have an account? Sign in here.

Sign In Now
×

• #### Activity

• Leaderboard
×
• Create New...

## Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.