Algebra.Com's Answer #824128 by ikleyn(52790)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Determine if the given set of vectors is linearly dependent or independent.
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document.write( "{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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document.write( "P3[R] is the linear space of polynomials of the degree 3 over the real number coefficients.\r\n" );
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document.write( "In this space, the vectors (the polynomials) \"1\" (constant terms), x, x^2 and x^3 form a basis.\r\n" );
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document.write( "The matrix of the transition from this basis to the given vectors is\r\n" );
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document.write( " A =  . (1)\r\n" );
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document.write( "To determine if the given vectors are linearly independent, it is enough to check if the matrix A (1) is non-degenerated.\r\n" );
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document.write( "In turn, it is enough to check if the determinant of the matrix A is non-zero real number.\r\n" );
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document.write( "There are different ways to do it. One of the ways is to use elementary transformations of the matrix (1).\r\n" );
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document.write( "To avoid this boring way, I used an online calculator www.reshish.com of the Internet open source.\r\n" );
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document.write( "I found that the determinant of the matrix (1) is -260.\r\n" );
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document.write( "The fact that the determinant is not zero means that the matrix A is non-degenerated.\r\n" );
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document.write( "In turn, it means that the given vectors are independent in the space P3[R].\r\n" );
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document.write( "Solved.\r
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