For those that are linearly dependent, express one of the polynomials as a linear combination of the others {1-2x,3x+x^2-x^3,1+x^2+2x^3,3+2x+3x^3}

Yulia

Yulia

Answered question

2021-09-03

Test the sets of polynomials for linear inde­pendence. For those that are linearly dependent, express one of the polynomials as a linear combination of the others {12x,3x+x2x3,1+x2+2x3,3+2x+3x3} in P3

Answer & Explanation

Cristiano Sears

Cristiano Sears

Skilled2021-09-04Added 96 answers

We have to determine whether or not we can find real numbers r, s,t, which are not all zero,
Such that
r(12x)+s(3x+x2x3)+t(1+x2+2x3)+q(3+2x+3x3)=0
To find all possible r, s,t we have to solve the augmented matrix equation:
[1013|02302|00110|00123|0]
R2R2+2R1
[1013|00328|00110|00123|0]
R3R2
[1013|00110|00328|00123|0]
R3R33R2 and R4R4+R2
[1013|00110|00018|00033|0]
R4R4+3R3
[1013|00110|00018|000027|0]
Hence,
r(12x)+s(3x+x2x3)+t(1+x2+2x3)+q(3+2x+3x3)=0
satisfies only when r=s=t=q=0
Therefore,
{(12x),(3x+x2x3),(1+x2+2x3),(3+2x+3x3)}
is a linearly independent set of polynomials.

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