Unique linear transformations

In summary: If you have a set of n linearly independent vectors in an n-dimensional space, any other vector in the space can be written as a linear combination of those n vectors. If you apply the linear transformation to that vector, and to each of the other n vectors, you get n vectors in the other space. Write those vectors as columns of a matrix. A little thought should convince you that that matrix is unique.In summary, the problem is to show that the linear transformation between two given vector spaces is unique, given a finite number of vectors in each space. To prove this, one approach is to use a system of equations and consider the matrix representation of the linear transformation. If a specific order is given for
  • #1
complexhuman
22
0
Unique linear transformations!

Problems agiain :cry: :cry: :cry:

Say I have 2 vector spaces with some finite number of vectors(can assume linear independency)...how can I show that the linear transformation between the two is unique?

Thanks in advance!
 
Physics news on Phys.org
  • #2
Hint: translate the problem using a system of equations.
 
  • #3
complexhuman said:
Problems agiain :cry: :cry: :cry:

Say I have 2 vector spaces with some finite number of vectors(can assume linear independency)...how can I show that the linear transformation between the two is unique?

Thanks in advance!

Perhaps I am misunderstanding something here. Do you mean that you are given a finite, linearly independent set of vectors in each space and the linear transformation must map each vector in one set into a corresponding vector in the other?

They way you have stated it, you can't prove the linear transformation is unique- it isn't. If the two sets are not bases for their vector spaces, then the linear transformation is not unique. Even if they are bases, unless you are requiring that the linear transformation map a specific vector in one set into a specific vector in the other then different linear transformations can map one set into the other, just rearranging which vector maps into which.

Assuming that you are given a basis for one space, in a specific order, and the linear transformation must map that into a basis for the other space, also in a given order, then you can show that the linear transformation is unique. One way would be to see what matrix represents that linear transformation in those bases.
 

Related to Unique linear transformations

What is a unique linear transformation?

A unique linear transformation is a function that maps vectors from one vector space to another, while preserving the linear structure. This means that the transformation must satisfy the properties of linearity, such as preserving addition and scalar multiplication.

How is a unique linear transformation different from a regular linear transformation?

A unique linear transformation is a linear transformation that is one-to-one and onto, meaning that each element in the output vector space has a unique corresponding input vector. This is not necessarily the case for regular linear transformations, which may map multiple input vectors to the same output vector.

What are some real-world applications of unique linear transformations?

Unique linear transformations can be used in various scientific and engineering fields, such as computer graphics, image processing, and data compression. They are also important in linear algebra, as they help in understanding the properties of vector spaces and their transformations.

How can I determine if a transformation is unique?

A transformation is unique if it satisfies the properties of linearity and is both one-to-one and onto. This can be determined by checking if the transformation preserves addition and scalar multiplication, and if each output vector has a unique corresponding input vector.

Are there any limitations to unique linear transformations?

One limitation of unique linear transformations is that they may not exist for all vector spaces. For example, if the dimensions of the input and output vector spaces are different, a unique linear transformation may not be possible. Additionally, the existence of a unique linear transformation also depends on the specific properties of the vector space and the transformation being considered.

Similar threads

I Want to understand how to express the derivative as a matrix
    • Linear and Abstract Algebra
Replies
8
Views
1K
I Analogy question for algebraists
    • Linear and Abstract Algebra
Replies
1
Views
984
B Commutative diagram conventions
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Dimension of a Linear Transformation Matrix
    • Linear and Abstract Algebra
Replies
4
Views
1K
MHB Mapping linear spaces to nonlinear ones.
    • Linear and Abstract Algebra
Replies
1
Views
1K
Vectors as geometric objects and vectors as any mathematical objects
    • Linear and Abstract Algebra
Replies
27
Views
1K
I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
970
I Given two linear transformations L and K, show ##K = \lambda L## holds
    • Linear and Abstract Algebra
Replies
9
Views
2K
I Linear Transformations: Why w1 is a Linear Combination of v
    • Linear and Abstract Algebra
Replies
6
Views
1K
I Change of Basis Matrix vs Transformation matrix in the same basis....
    • Linear and Abstract Algebra
Replies
12
Views
3K

Hot Threads

Recent Insights

Back
Top