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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 ?

Thanks in advance

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  • $\begingroup$ Welcome to Maths SX! Hint: What is the dimension of the set of polynomials of degree $\le 2$? $\endgroup$
    – Bernard
    Jan 19, 2020 at 17:47
  • $\begingroup$ 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). $\endgroup$ Jan 19, 2020 at 17:50
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    $\begingroup$ Yes sorry if I forgot the field , it's R $\endgroup$
    – Hustler885
    Jan 19, 2020 at 17:55

2 Answers 2

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Consider mapping this set to $\mathbb{R^3}$

that is $f(ax^{k})=a \cdot e_{k+1}$ $(\text{for example} \ \ f(\pi x^{0}) =\pi \cdot e_1$

This is helpful in that it converts these polynomials to columns of a matrix that you can simply reduce to check linear independence

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Let the polynomials be numbered $p_1,...,p_4$. Note that $$\frac{\sqrt3}{\sqrt2}\cdot p_2-p_3-\frac{3+e}\pi p_4=0$$

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