Understanding Linear Transformations and Their Applications in Vector Spaces

L(a v) = a L(v) (it doesn't matter if you first multiply by a scalar and then apply the linear operation or first apply the linear operation and then multiply by a scalar).In summary, a linear transformation is a function that takes a vector from one vector space to another and respects the operations of addition and scalar multiplication. It can be thought of as a matrix, with the rows of the matrix representing the constants of the linear combination of vectors in the basis of the second vector space. Transforming a vector from one vector space to another doesn't give you a matrix, but is the same as operating on that vector with a matrix.
  • #1
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?
 
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  • #2
Take a linear map from V to U. If V has basis [tex]{v_1, v_2,..., v_n}[/tex] and U has basis [tex]{u_1, u_2,..., u_m}[/tex]

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

It should be obvious why [tex]T(v_k)[/tex] 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)
 
  • #3
Office_Shredder said:
Take a linear map from V to U. If V has basis [tex]{v_1, v_2,..., v_n}[/tex] and U has basis [tex]{u_1, u_2,..., u_m}[/tex]

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

It should be obvious why [tex]T(v_k)[/tex] 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.
[tex]T(v_k)=a_1w_1+a_2w_2...a_nw_n[/tex] 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.
 
  • #4
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 wi's for each T(vk).

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
 
  • #5
Office_Shredder said:
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 wi's for each T(vk).

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?
 
  • #6
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
 
  • #7
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).
 
  • #8
...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.)
 
  • #9
Well, I do not get how you actually transform a vector from one vectorspace to another so maybe an example would help.
 

FAQ: Understanding Linear Transformations and Their Applications in Vector Spaces

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the linear structure of the original space. This means that the transformation preserves addition and scalar multiplication properties.

How are linear transformations represented?

Linear transformations can be represented using matrices or linear equations. Matrices provide a compact and efficient way to represent the transformation, while linear equations show the relationship between the input and output vectors.

What are some common applications of linear transformations?

Linear transformations have many applications in fields such as physics, engineering, computer graphics, and machine learning. They are used to solve systems of linear equations, model physical systems, compress and decompress images, and transform data in machine learning algorithms.

How do you determine if a transformation is linear?

To determine if a transformation is linear, you can check if it satisfies two properties: additivity and homogeneity. Additivity means that the transformation of the sum of two vectors is equal to the sum of their individual transformations. Homogeneity means that the transformation of a vector multiplied by a scalar is equal to the scalar multiplied by the transformation of the original vector.

Can linear transformations be applied to any vector space?

No, linear transformations can only be applied to vector spaces that have the same dimension. This means that the transformation must have the same number of inputs and outputs. Additionally, the vector space must also satisfy the properties of additivity and homogeneity for the transformation to be valid.

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