Linear Transformations and Dual Vectors: Understanding Matrix Representations

In summary, the conversation discusses the construction of matrix representations for linear transformations between vector spaces, and the relationship between the transpose of a matrix and the dual space of the original vector space. The question is whether there exist bases on the dual spaces that make the statements true. The conversation also touches on the concept of dual bases and the relationship between a basis for a vector space and its dual space.
  • #1
mhazelm
41
0
It's been so long since I thought about this, I just need to know if this is correct.

If I have the matrix representation of a linear transformation between vector spaces V and W, and I take the transpose of the matrix, am I in essence constructing the matrix representation of a corresponding transformation from W* to V* (where * denotes dual space)?

And if I take the transpose of a row vector in V, can I think of the resulting column vector as an element of V*?

Thanks in advance! :smile:
 
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  • #2
When we talk about matrices, we are implicitly fixing a basis for the spaces under consideration. So, let's rephrase your question: are there bases on W* and V* that make what you said true?

What do you think?
 
  • #3
Given a basis on W, there exist a corresponding basis on W*. Do you know what it is?
 
  • #4
I thought that given any vector, we could always find the vector dual to it... so couldn't we just find the vectors dual to our basis vectors and call that our basis for W*?

I think in fixing the basis for V we fix the basis for V*. Since they're isomorphic, don't we kind of get the basis for V* for "free"?
 
  • #5
Yes, that's what I just said. Given a basis for V, what is the dual basis?
 

FAQ: Linear Transformations and Dual Vectors: Understanding Matrix Representations

What is a linear transformation and how is it represented by a matrix?

A linear transformation is a mathematical function that maps one vector space to another in a linear manner. It is represented by a matrix by assigning a column vector to each basis vector of the input vector space and transforming it into a new basis vector using matrix multiplication.

What are dual vectors and how are they related to linear transformations?

Dual vectors are also known as covectors or one-forms, and they are linear functionals that map a vector to a scalar. They are related to linear transformations because they can be used to represent linear transformations as well. The matrix representation of a dual vector is the transpose of the matrix representation of the linear transformation it represents.

How do we interpret the entries of a matrix representation of a linear transformation?

The entries of a matrix representation of a linear transformation represent the coefficients that determine the transformation of the basis vectors in the input vector space to the basis vectors in the output vector space. These coefficients can be used to calculate the transformation of any vector in the input space.

Can a linear transformation have multiple matrix representations?

Yes, a linear transformation can have multiple matrix representations depending on the choice of basis vectors for the input and output vector spaces. However, all these representations are equivalent and can be transformed into one another using a change of basis matrix.

How can understanding matrix representations of linear transformations be useful in real-world applications?

Matrix representations of linear transformations are useful in many real-world applications, such as computer graphics, data compression, and machine learning. They allow us to efficiently perform calculations and transformations on large datasets and simplify complex mathematical operations. They also provide a powerful tool for analyzing and understanding the behavior of systems in various fields of science and engineering.

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