Linear Transformations and Dual Vectors: Understanding Matrix Representations

[mhazelm]

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!

 

[morphism]

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?

 

[HallsofIvy]

Given a basis on W, there exist a corresponding basis on W*. Do you know what it is?

 

[mhazelm]

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"?

 

[HallsofIvy]

Yes, that's what I just said. Given a basis for V, what is the dual basis?