1.5 - Linear Algebra-115 Solutions to Third Homework...

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Linear Algebra -115 Solutions to Third Homework Problem 2( d, f ) (Section 1 . 5) (d) Assume that a ( x 3 - x )+ b (2 x 2 +4)+ c ( - 2 x 3 +3 x 2 +2 x +6) = 0. Doing some algebra we have ( a - 2 c ) x 3 +(2 b +3 c ) x 2 +( - a +2 c ) x +(4 b +6 c ) = 0. This gives rise to the system with corresponding matrix 1 0 - 2 0 0 2 3 0 - 1 0 2 0 0 4 6 0 . We row-reduce the matrix: 1 0 - 2 0 0 2 3 0 - 1 0 2 0 0 4 6 0 1 0 - 2 0 0 2 3 0 0 0 0 0 0 4 6 0 1 0 - 2 0 0 2 3 0 0 0 0 0 0 0 0 0 , which means that the original system has non-zero solutions, or that the polynomials aren’t linearly independent. (f) To test given vector for linear independence we assemble them into a matrix. Then we row-reduce the matrix: 1 2 - 1 - 1 0 2 2 1 - 1 1 2 - 1 0 2 1 0 - 3 1 1 2 - 1 0 1 1 / 2 0 - 3 1 1 2 - 1 0 1 1 / 2 0 0 5 / 2 1 2 - 1 0 1 1 / 2 0 0 1 , The resulting matrix is invertible. This implies that the matrix we started with is invertible and so the three vectors are linearly independent.
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