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PostPosted: Fri, 4 Nov 2011 03:57:38 UTC 
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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?


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PostPosted: Fri, 4 Nov 2011 05:52:27 UTC 
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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.

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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PostPosted: Fri, 4 Nov 2011 06:05:01 UTC 
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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. . .

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