How to solve this linear algebra problem?

Express the polynomial p(t)= 3+9t^2 as a linear combination of the polynomials:

f(t)=1+2t^2

g(t)=1+t

h(t)=1+2t+t^2.

Please explain...

Thanks!!

1 Answer

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  • 9 years ago
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    So there must be some c1, c2, c3 such that

    c1*f + c2*g + c3*h = p

    c1(1 + 2t^2) + c2(1 + t) + c3(1 + 2t + t^2) = 3 + 9t^2

    Now collect all terms.

    (c1 + c2 + c3) + (c2 + 2c3)t + (2c1 + c3)t^2 = 3 + 9t^2

    So now you have 3 equations and 3 unknowns. You can solve this via matrices but this seems straightforward to solve via substitution.

    Matching all the coefficients you get

    c1 + c2 + c3 = 3

    c2 + 2c3 = 0 --> c2 = -2c3

    2c1 + c3 = 9 --> c1 = (9 - c3)/2

    Substituing you get

    (9 - c3)/2 - 2c3 + c3 = 3

    4.5 - 3c3/2 = 3

    9 - 3c3 = 6

    c3 = 1

    c2 = -2

    c1 = 4

    So now go back to the original equation and substitue the coefficients.

    4f(t) - 2g(t) + h(t) = p(t)

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