Understanding Linear Transformations and Their Applications in Vector Spaces

[lo2]

Ok I have to do this Linear Algebra 'Report', it is not really a Report, Report was just the best I could come up with to describe it. But anyway I have read about Vector spaces and basics and I think that I get it. Then I started reading about Linear transformation and I think it is a bit weird so therefore I would like to ask you for help.

As far as I have understood it is when you want to, transform a linear combination from one vector space to another vector space, and in order to make the same linear combination you have to change it so that it fits into the new vector space. Is this just plain crap or is it right?

But I would to get an example of a so called transformation. You use two 'laws' f(u+v)=f(u)+f(v) and f(tu)=tf(u) (the bold mean that it is a vector) but I do not really see how these two laws can help you?

And furthermore sometimes you must 'loose' a dimension? And can you transform a linear combination to a vector space with more dimension than the first vector space. For example could you transform something from R^2 to R^3?

 

[Office_Shredder]

Take a linear map from V to U. If V has basis $${v_1, v_2,..., v_n}$$ and U has basis $${u_1, u_2,..., u_m}$$

then you can say $$T(v_k) = \sum{a_iu_i}$$ for each k from 1 to n, and this defines a unique linear transformation.

It should be obvious why $$T(v_k)$$ can be written as a unique linear combination of the basis of U. Why this makes T unique is a bit more in depth (and showing that each linear combination can be written this way is too)

 

[lo2]

Take a linear map from V to U. If V has basis $${v_1, v_2,..., v_n}$$ and U has basis $${u_1, u_2,..., u_m}$$

then you can say $$T(v_k) = \sum{a_iu_i}$$ for each k from 1 to n, and this defines a unique linear transformation.

It should be obvious why $$T(v_k)$$ can be written as a unique linear combination of the basis of U. Why this makes T unique is a bit more in depth (and showing that each linear combination can be written this way is too)

So for a vector space V where you have a basis the number depends on the dimension and then you have also got a vector space W for which the same applies.

Then what you do is that the linear transformation of V to W is.
$$T(v_k)=a_1w_1+a_2w_2...a_nw_n$$ am I right?
But then what is T(x) and what is this Linear transformation could someone give me a brief explanation/definiton. On what to think of it as.

 

[Office_Shredder]

Linear transformations can be thought of as matrices. It's provable there is a correspondence between matrices acting on coordinate vectors of a given vector, and matrices acting on those coordinates. So just think of linear transformations as matrices.

What you get is that the rows of the matrix corresponding to T are the constants of the w_i's for each T(v_k).

So mapping from an n dimensional space to an m dimensional space gives you an mxn matrix of constants. If you know a basis of each space, you can calculate the constants as described above. It should be noted that given different bases, your matrix will be different

 

[lo2]

Linear transformations can be thought of as matrices. It's provable there is a correspondence between matrices acting on coordinate vectors of a given vector, and matrices acting on those coordinates. So just think of linear transformations as matrices.

What you get is that the rows of the matrix corresponding to T are the constants of the w_i's for each T(v_k).

So mapping from an n dimensional space to an m dimensional space gives you an mxn matrix of constants. If you know a basis of each space, you can calculate the constants as described above. It should be noted that given different bases, your matrix will be different

So when you transform a linear combination from one vector space to another vector space, you get a matrix?

 

[Office_Shredder]

Not really. Take a linear transformation T:V -> U

If you take T(v) for each v in the basis of V (a specific basis, it's going to be different for each basis chosen), you're going to get a linear combination of the vectors in the basis of U. If you know what all those linear combinations are, you can then find T(v') for any v' in V (write v' as a linear combination of the elements in the basis of V, use the linearity of T and you can break it up into things you already know). This defines a unique linear transformation.

So what you can do, is say v' is written as a linear combination of the vectors in the basis of V. You can take the coefficients of those vectors, and make a column vector out of those scalars. You call this the coordinate vector of v' Associated with each linear transformation is a unique matrix that takes the coordinate vector of v' to the coordinate vector of T(v') (which is the scalars of the linear combination of vectors in the basis of U).

So transforming a vector from one vector space into a vector of another doesn't give you a matrix, but is the same as operating on that vector with a matrix

 

[HallsofIvy]

You keep talking about "transforming a linear combination". You really just mean "vector" rather than "linear combination". Since the only two operations on a vector space are adding vectors are multiplying vectors by scalars: a "linear combination" is just a general combination of those two operations. Any vector can be written as a linear combination of other vectors. A "linear transformation" is a function from one vector space to another that "respects" those operations: L(u+ v)= L(u)+ L(v) (it doesn't matter if first add and then apply the linear operation or first apply the linear transformation and then add) and L(av)= aL(v) (it doesn't matter if you first multiply by a scalar and then apply the linear transformation or first apply the linear transformation and then multiply by the scalar).

 

[radou]

...or you can just require one equvalent condition - that a linear transformation must satisfy L(au + bv)=aL(u) + bL(v), where a and b are scalars, and u and v are vectors, of course. (Probably wasn't necessary to point this out, but nevermind.)

 

[lo2]

Well, I do not get how you actually transform a vector from one vectorspace to another so maybe an example would help.