My text is not helping at all and the wikipedia explanations of the concepts: basis and dimension are way over my head.

Let V be the space of polynomials in x and y of degree <= 10. Specify a basis of V and compute dimV.

thanks

jblorien

Printable View

- Mar 15th 2008, 08:13 AMjblorienDimension and basis
My text is not helping at all and the wikipedia explanations of the concepts: basis and dimension are way over my head.

Let V be the space of polynomials in x and y of degree <= 10. Specify a basis of V and compute dimV.

thanks

jblorien - Mar 15th 2008, 09:17 AMgalactus
To be a basis it must span and be linearly independent.

The polynomials $\displaystyle 1,x,x^{2},x^{3},.....,x^{10}$ span the vector

space of $\displaystyle P_{n}$ since each p in $\displaystyle P_{10}$ can be

written as $\displaystyle p=a_{0}+a_{1}x+....+a_{10}x^{10}$ which is a

linear combination of $\displaystyle S=\left[1,x,x^{2},.....,x^{10}\right]$.

$\displaystyle dim(P_{n})=n+1$. The basis has n+1 vectors. In other words, it has dimension 11.