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?

Correct option is D

Solution:

​Basis of $\mathbb{R}$-vector space of all polynomials in two variables over $\mathbb{R}$ having a total degree of at most $l$ will be:

$$B = \{ 1, x, y, x^2, xy, y^2, \dots, x^l, x^{l-1}y, x^{l-2}y^2, \dots, y^l \}$$

So in basis:
Number of vectors of degree 0: = 1

Number of vectors of degree 1: = 2

​​Number of vectors of degree 2: = 3

.

.

.

Number of vectors of degree l: = l+1

So, dimension of vector space =Number of vectors in basis B= $$1 + 2 + 3 + \dots + (l+1)$$

$$1 + 2 + 3 + \dots + (l+1) = \frac{(l+1)(l+2)}{2}$$​​​