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Determine if the given set of vectors is linearly dependent or independent.
{1+x^2 , -1-3x+4x^2+5x^3 , 2+5x-6x^3 , 4+6x+3x^2+7x^3} in P3[R].
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P3[R] is the linear space of polynomials of the degree 3 over the real number coefficients.
In this space, the vectors (the polynomials) "1" (constant terms), x, x^2 and x^3 form a basis.
The matrix of the transition from this basis to the given vectors is
A = . (1)
To determine if the given vectors are linearly independent, it is enough to check if the matrix A (1) is non-degenerated.
In turn, it is enough to check if the determinant of the matrix A is non-zero real number.
There are different ways to do it. One of the ways is to use elementary transformations of the matrix (1).
To avoid this boring way, I used an online calculator www.reshish.com of the Internet open source.
I found that the determinant of the matrix (1) is -260.
The fact that the determinant is not zero means that the matrix A is non-degenerated.
In turn, it means that the given vectors are independent in the space P3[R].
Solved.