# Check whether a set of polynomials is linearly independent.

I have to find if this set of polynomial is linear dependent or independent $$S=\{x^2−1,\;\sqrt2x+\sqrt3,\;\sqrt3x−e,\;\pi\}$$

I know it is dependent given that is a 4 elements subset of P2 that can contain at maximum 3 independent elements. However I am struggling at proving it with the definition of linear independence and at identifying the element to remove to make it an independent set.Could someone help me ?

• Welcome to Maths SX! Hint: What is the dimension of the set of polynomials of degree $\le 2$? Jan 19, 2020 at 17:47
• It's very important what field you're asking about linear independence over; the answer is different over $\Bbb Q$ and over $\Bbb R$ (it seems that the latter is what you mean). Jan 19, 2020 at 17:50
• Yes sorry if I forgot the field , it's R Jan 19, 2020 at 17:55

Consider mapping this set to $$\mathbb{R^3}$$
that is $$f(ax^{k})=a \cdot e_{k+1}$$ $$(\text{for example} \ \ f(\pi x^{0}) =\pi \cdot e_1$$
Let the polynomials be numbered $$p_1,...,p_4$$. Note that $$\frac{\sqrt3}{\sqrt2}\cdot p_2-p_3-\frac{3+e}\pi p_4=0$$