Dimension and basis

**jblorien** · March 15th 2008, 08:13 AM

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

**galactus** · March 15th 2008, 09:17 AM

To be a basis it must span and be linearly independent.

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

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

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

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

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

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