Linear Transformations and Bases

[Math Amateur]

I need some help or at least some assurance that my thinking on linear transformations and their matrix representations is correct.

I assume when we specify a linear transformation eg F(x,y, z) = (3x +y, y+z, 2x-3z) for example, that this is specified by its action on the variables and is not with respect to any basis.

However when we specify the matrix of a linear transformation T: V --> W that this is with respect to a basis in V and a basis in W

Of course if we have a linear transformation S: V -->V it could be that the two bases are the same.

If no basis is mentioned regarding the matrix of a linear transformation, then I am assuming the standard bases are assumed.

Can someone either confirm I am correct in my thinking or point out the errors in my thinking?

Peter

 

[Office_Shredder]

If no basis is mentioned regarding the matrix of a linear transformation, then I am assuming the standard bases are assumed.

What is the standard basis of a generic vector space V?

 

[Math Amateur]

Yes, good point ... I guess I should have specified Euclidean space for that assumption to make sense ...

Are my other assumptions/interpretations OK?

 

[Office_Shredder]

Everything is correct, but I think you have a bit of a hangup on how linear transformations on vector spaces are described

I assume when we specify a linear transformation eg F(x,y, z) = (3x +y, y+z, 2x-3z) for example, that this is specified by its action on the variables and is not with respect to any basis.

However when we specify the matrix of a linear transformation T: V --> W that this is with respect to a basis in V and a basis in W

The key here is that in your first example you specified a linear transformation on R³ without defining it as a matrix multiplication, so no basis is required and no matrix is ever constructed. If you wanted to define F as a matrix multiplication you would need to specify a basis of R³ - on Euclidean space this step is often omitted because everyone assumes your basis is the standard basis (1,0,0),(0,1,0),(0,0,1) (and trying to specify F as matrix multiplication with respect to a different basis is literally just extra work).

If we want to specify a linear transformation V--> W as a matrix multiplication we need to pick bases of V and W to identify them with Euclidean space. But we can have linear transformations that are not represented as matrix multiplications. For example let V be the set of all polynomials of degree <= 3 (this is a 4 dimensional vector space over R) and let W be R. Then consider
$$I:V\to \mathbb{R},\ I(p(x)) = \int_0^1 p(x) dx$$
I is a linear transformation and I never specified a basis for V in order to tell you the function because I didn't tell you what I was as a matrix multiplication

 

[Math Amateur]

Thanks so much for that post - most helpful

I suppose the essential thing needed to be able to derive a matrix of a linear transformation is that the vector spaces involved need to be over a field F where F = R or C.

Is that correct?

Missed your example due to some latex error or other - pity - would have like to have viewed your example

Thanks again!

Peter