# Determine whether the following polynomials u,v, w in P(t) are linearly dependent or independent:u=t^3-4t^2+3t+3,v=t^3+2t^2+4t-1,w=2t^3-t^2-3t+5

Determine whether the following polynomials u,v, w in P(t) are linearly dependent or independent:
$$\displaystyle{u}={t}^{{3}}-{4}{t}^{{2}}+{3}{t}+{3},$$

$${v}={t}^{{3}}+{2}{t}^{{2}}+{4}{t}-{1},$$

$${w}={2}{t}^{{3}}-{t}^{{2}}-{3}{t}+{5}$$

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Let $$\displaystyle{c}_{{1}},{c}_{{2}},{c}_{{3}}$$ be the constants.
So $$\displaystyle{u}{c}_{{1}}+{v}{c}_{{2}}+{w}{c}_{{3}}={0}$$ gives
$$\displaystyle{c}_{{1}}{\left({t}^{{3}}-{4}{t}^{{2}}+{3}{t}+{3}\right)}+{c}_{{2}}{\left({t}^{{3}}+{2}{t}^{{2}}+{4}{t}-{1}\right)}+{c}_{{3}}{\left({2}{t}^{{3}}-{t}^{{2}}={3}{t}+{5}\right)}={0}$$
$$\displaystyle{t}^{{3}}{\left({c}_{{1}}+{c}_{{2}}+{2}{c}_{{3}}\right)}+{t}^{{2}}{\left(-{4}{c}_{{1}}+{2}{c}_{{2}}-{c}_{{3}}\right)}+{t}{\left({3}{c}_{{1}}+{4}{c}_{{2}}-{3}{c}_{{3}}\right)}+{\left({3}{x}_{{1}}-{c}_{{2}}+{5}{c}_{{3}}\right)}={0}$$
Comparing coefficients of $$\displaystyle{t}^{{3}},{t}^{{2}},{t}$$ and constant term both sides we get,
$$\displaystyle{c}_{{1}}+{c}_{{2}}+{2}{c}_{{3}}={0}$$ (1)
$$\displaystyle-{4}{c}_{{1}}+{2}{c}_{{2}}-{c}_{{3}}={0}$$ (2)
$$\displaystyle{3}{c}_{{1}}+{4}{c}_{{2}}-{3}{c}_{{3}}=$$ (3)
and $$\displaystyle{3}{c}_{{1}}-{c}_{{2}}+{5}{c}_{{3}}={0}$$ (4)
Subtracting equation (4) from equation (3), we get
$$\displaystyle{8}{c}_{{3}}={5}{c}_{{2}}$$
$$\displaystyle{c}_{{3}}={\left(\frac{{5}}{{8}}\right)}{c}_{{2}}$$ (5)
Substituting the value of $$c_3$$ in equation (1), we get
$$\displaystyle{c}_{{1}}+{c}_{{2}}+{2}{\left(\frac{{5}}{{8}}\right)}{c}_{{2}}={0}$$
$$\displaystyle{c}_{{1}}={\left(-\frac{{9}}{{4}}\right)}{c}_{{2}}$$ (6)
Substituting values from (5) & (6) in equation (1), we get
$$\displaystyle{\left(-\frac{{9}}{{4}}\right)}{c}_{{2}}+{c}_{{2}}+{\left(\frac{{5}}{{8}}\right)}{c}_{{2}}={0}$$
So, $$\displaystyle{c}_{{2}}={0}$$
And so from (5) and (6) $$\displaystyle{c}_{{1}}={c}_{{2}}={c}_{{3}}={0}$$
Therefore, polynomials $$\displaystyle{u}={t}^{{3}}-{4}{t}^{{2}}+{3}{t}+{3},{v}={t}^{{3}}+{2}{t}^{{2}}+{4}{t}-{1},{w}={2}{t}^{{3}}-{t}^{{2}}-{3}{t}+{5}$$ are linearly independent.