# Vectors, dual vectors

1. Sep 14, 2008

### 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*?

2. Sep 16, 2008

### 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?

3. Sep 17, 2008

### HallsofIvy

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

4. Sep 19, 2008

### 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"?

5. Sep 19, 2008

### HallsofIvy

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