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Let l ≥ 1 be a positive integer. What is the dimension of the ℝ-vector space of all polynomials in two variables over R having a total degree of at most l?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. l + 1
  2. l(l − 1)
  3. l(l + 1)/2
  4. (l + 1)(l + 2)/2

Answer (Detailed Solution Below)

Option 4 : (l + 1)(l + 2)/2
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Detailed Solution

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Concept:

Dimension of the vector space of polynomial f of m variable having degree at most n is \(^{m+n}C_m\)

Explanation:

Here number of variable of polynomial is 2 and degree is at most l so

dimension = \(^{l+2}C_2\) = \(\frac{(l+2)!}{2!\times l!}\) = \(\frac{(l+2)(l+1)\times l!}{2\times l!}\) = (l + 1)(l + 2)/2

Option (4) is correct

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