Vector Space Help: Understanding Notation & Polynomials

[elle]

Vector space help please..

Hi,
Just started a linear algebra course recently but I am confused with the notation used

http://i9.tinypic.com/2w4za50.jpg

I am unsure how to proceed with this question. Can someone help? The part highlighted, what does it mean? 2x2 matrix of P? The P represents the polynomial entries? Can give me an example to give me a head start? Many thanks!

 

[radou]

Yes, it is the set of all 2x2 matrices whose elements are real polynomials. Start with: http://mathworld.wolfram.com/VectorSpace.html" .

 

[elle]

Yes, it is the set of all 2x2 matrices whose elements are real polynomials. Start with: http://mathworld.wolfram.com/VectorSpace.html" .

Thanks

Is it something like this?

http://i10.tinypic.com/48qhc1w.jpg

 

[radou]

Thanks

Is it something like this?

http://i10.tinypic.com/48qhc1w.jpg

Yes, it's exactly something like this. Now just look at the definition of a vector space and at the properties that the addition and scalar multiplication must satisfy to proove if it's a vector space or not.

 

[elle]

Yes, it's exactly something like this. Now just look at the definition of a vector space and at the properties that the addition and scalar multiplication must satisfy to proove if it's a vector space or not.

thanks for the confirmation again

Ok hmm I don't know if I'm on the right track but do I have to have two different matrices? Let's say matrix A and matrix B where A has elements:

http://i10.tinypic.com/4g7c4lj.jpg

and B with similar elements in order to check whether they satisfy closure by addition and multiplication? Or have I interpreted the definition totally wrong?

 

[radou]

thanks for the confirmation again

Ok hmm I don't know if I'm on the right track but do I have to have two different matrices? Let's say matrix A and matrix B where A has elements:

http://i10.tinypic.com/4g7c4lj.jpg

and B with similar elements in order to check whether they satisfy closure by addition and multiplication? Or have I interpreted the definition totally wrong?

You're on the right track.

 

[elle]

You're on the right track.

Thanks

I've just noticed that I've chosen specific polynomials for my matrix entries so is that wrong? what would the matrix look like with general polynomial entries if a degree isn't given? Hmm am i making any sense here

 

[radou]

Thanks

I've just noticed that I've chosen specific polynomials for my matrix entries so is that wrong? what would the matrix look like with general polynomial entries if a degree isn't given? Hmm am i making any sense here

You don't have to write down specific polynomials as entries in your matrix. It is enough to write down something like $$\left(\begin{array}{cc}p_{1} & p_{2}\\p_{3} & p_{4}\end{array}\right)$$, where $$p_{i}, i = 1, \cdots, 4$$ are your real polynomials, which is the only thing that matters, unlike their degrees.