a) Find a basis for the span of the polynomials. {1-4x2, 3x+4x, 2x, 3-4x+2x) b) Find...

  

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a) Find a basis for the span of the polynomials. {1-4x2, 3x+4x, 2x, 3-4x+2x) b) Find the coordinates of the polynomial in the basis found in part a) p(x)=5x+2x-3 a) Find a basis for the span of the polynomials. {1-4x2, 3x+4x, 2x, 3-4x+2x) b) Find the coordinates of the polynomial in the basis found in part a) p(x)=5x+2x-3

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a To find a basis for the span of the given set of polynomials we need to determine which polynomials are linearly independent and can be used to form a basis Lets denote the polynomials as follows px 1 4x px 3x 4x px 2x px 3 4x 2x We can write the polynomials as vectors with their coefficients as components px 1 0 4 px 0 3 4 px 0 2 0 px 3 4 2 To find a basis we need to check if these vectors are linearly independent We can do this by row reducing the augmented matrix formed by these vectors and checking for linear dependence The augmented matrix is 1 0 0 3 0 3 2 4 4 4 0 2 Row reducing this matrix we get 1 0 0 3 0 1 0 2 0 0 1 1 Since the rowreduced form of the matrix has three nonzero rows these vectors are linearly independent Therefore a basis for the span of the polynomials 1 4x 3x 4x 2x 3 4x 2x is 1 4x 3x 4x 2x b To find the coordinates of the polynomial ... View the full answer