How Do You Calculate VμVμ with a Metric Tensor?

[andrey21]

Consider the vector V_μ(3,1)
Find V^μV_μ




Now here is my attempt

Using the following:

V_μ=g_μvV^v

I could calculate:
V^v=(3,1)

But how can I now manipulate this to obtain V^μ

 

[andrey21]

Using the fact that:

g_μv= 2x2 identity matrixand the dot product is given by:

V^μV_μ=g_μv V^μV^v

Therefore:

V^μ=g^μvV_v

So is it correct to say:

V^μ= (3,1)

 

[Mark44]

You really need to give some context here, and enlighten us on what the notation means. For example, what does V_μ(3, 1) mean? Also, how is V_μ different from V^μ? Are you just being sloppy with subscripts and superscripts?
Why are you using convoluted notation such as g_μv for the 2x2 identity matrix, when I₂ is much clearer?

I suspect that μ and v might be bases, but nowhere in your problem description does it say what these are.

 

[andrey21]

Im told to consider the specific example of the vector:

V_μ = (3,1) in the Cartesian coordinates.

g_μv is the metric tensor

Yes I believe they are bases, the question is based around raising and lowering the index in tensors