Use the inner product p, q = a0b0 + a1b1 + a2b2 to find p, q, p, q, and d(p, q) for the polynomials in P2.p(x) = 5 + x2, q(x) = 5 − x2(a) p, q (b) p (c) q (d) d(p, q)
Question
Use the inner product p, q = a0b0 + a1b1 + a2b2 to find p, q, p, q, and d(p, q) for the polynomials in P2.p(x) = 5 + x2, q(x) = 5 − x2(a) p, q (b) p (c) q (d) d(p, q)
Solution 1
(a) To find the inner product of p and q, we need to multiply the coefficients of the corresponding terms in p and q and then add them up. The polynomials p(x) = 5 + x^2 and q(x) = 5 - x^2 have coefficients a0 = 5, a2 = 1 for p and b0 = 5, b2 = -1 for q (note that the coefficients of x are zero since there are no x terms in the polynomials). So, the inner product p, q = a0b0 + a1b1 + a2b2 = 55 + 00 + 1*(-1) = 25 - 1 = 24.
(b) The inner product of p with itself is p, p = a0a0 + a1a1 + a2a2 = 55 + 00 + 11 = 25 + 0 + 1 = 26.
(c) The inner product of q with itself is q, q = b0b0 + b1b1 + b2b2 = 55 + 00 + (-1)(-1) = 25 + 0 + 1 = 26.
(d) The distance between p and q is given by the square root of the inner product of (p-q) with itself. The polynomial p-q = (5 + x^2) - (5 - x^2) = 2x^2. The coefficients of p-q are c0 = 0, c1 = 0, c2 = 2. So, d(p, q) = sqrt(c0c0 + c1c1 + c2c2) = sqrt(00 + 00 + 22) = sqrt(4) = 2.
Solution 2
(a) To find the inner product of p and q, we need to multiply the coefficients of the corresponding terms in p and q and then add them up. The polynomials p(x) = 5 + x^2 and q(x) = 5 - x^2 have coefficients a0 = 5, a2 = 1 for p and b0 = 5, b2 = -1 for q (note that the coefficients of x are zero since there are no x terms in the polynomials). So, the inner product p, q = a0b0 + a1b1 + a2b2 = 55 + 00 + 1*(-1) = 25 - 1 = 24.
(b) The inner product of p with itself is p, p = a0a0 + a1a1 + a2a2 = 55 + 00 + 11 = 25 + 0 + 1 = 26.
(c) The inner product of q with itself is q, q = b0b0 + b1b1 + b2b2 = 55 + 00 + (-1)(-1) = 25 + 0 + 1 = 26.
(d) The distance between p and q is given by the square root of the inner product of (p-q) with itself. The polynomial p-q = (5 + x^2) - (5 - x^2) = 2x^2. The coefficients of p-q are c0 = 0, c1 = 0, c2 = 2. So, d(p, q) = sqrt(c0c0 + c1c1 + c2c2) = sqrt(00 + 00 + 22) = sqrt(4) = 2.
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