Dimension and basis of a vector space

goteamusa · **Joined:** Fri, 30 Oct 2009 16:33:10 UTC **Posts:** 261

The question is: Find the dimension of the vector space V and give a basis for V. V={p(x) in P2:p(0)=0}

Well since this is P2, I know that the exponent of the polynomial must not exceed 2. So I am looking at polynomials of the form $\ ax^{2}+bx+c$ but because of the initial condition I know c=0. So what would the dimension be - $\ ax^{2}+bx$ ?
Also I know that the conditions for a basis are 1. that the vectors span S and 2. it is linearly independent. I am familiar with doing this for vectors, but I have never had to find a basis for a polynomial. What do I need to do to?

outermeasure · **Posted:** Fri, 4 Nov 2011 05:52:27 UTC

goteamusa wrote:

The question is: Find the dimension of the vector space V and give a basis for V. V={p(x) in P2:p(0)=0}

Well since this is P2, I know that the exponent of the polynomial must not exceed 2. So I am looking at polynomials of the form $\ ax^{2}+bx+c$ but because of the initial condition I know c=0. So what would the dimension be - $\ ax^{2}+bx$ ?
Also I know that the conditions for a basis are 1. that the vectors span S and 2. it is linearly independent. I am familiar with doing this for vectors, but I have never had to find a basis for a polynomial. What do I need to do to?

Not much differences --- individual polynomials are your vectors, and the space of polynomials of degree at most 2 is your vector space.

Shadow · **Posted:** Fri, 4 Nov 2011 06:05:01 UTC

goteamusa wrote:

The question is: Find the dimension of the vector space V and give a basis for V. V={p(x) in P2:p(0)=0}

Well since this is P2, I know that the exponent of the polynomial must not exceed 2. So I am looking at polynomials of the form $\ ax^{2}+bx+c$ but because of the initial condition I know c=0. So what would the dimension be - $\ ax^{2}+bx$ ?
Also I know that the conditions for a basis are 1. that the vectors span S and 2. it is linearly independent. I am familiar with doing this for vectors, but I have never had to find a basis for a polynomial. What do I need to do to?

Eh? Use the coordinate mapping to look at $P_2\cong \mathbb{R}^3$ , then this is the space of vectors which look like $\begin{pmatrix}x \\ y \\ 0\end{pmatrix}$ so. . .

S.O.S. Mathematics CyberBoard

Dimension and basis of a vector space

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