Using the inner product (,): P₂ (R) × P₂ (R) → R, (p, q) = p(0)q(0) + p(1)q(1) + p(2)q(2) calculate (10x - 2x², 1 – 5x – 2x²).
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- Find the general formula for T: R3 -> R3, if T(1,0,1) = (1,2,0) , T(1,-2,1) = (0,1,0) and T (0,0,1) = (0,2, -1). Also find T(2,-3,1).Consider the polynomials a(x) = x' + x³ + x + 4 and b(x) = 2r³ + 4.x² + x + 2 from Z5[r]. a) Find ged(a(x), b(x)). b) Express gcd(a(x), b(x)) as a linear combination of a(x) and b(x). c) Express b(x) as a product of irreducible factors. Explain why each factor is irreducible.h(x1,x2)=x2e-x1 at (2,-1)
- ce #3 Consider p(x) = 3x - 2 and q (x) = x + 7. The evaluation inner product is defined as (p, q) = p (x1) a (x₁) +p (x₂) q (x₂)+p(x3) q (x3). For (1, 2, 3) = (1, -1, 3), what is the distance d (p, q)? O √179 C E 21 84 803 54 $ 4 DEC R G Search or type URL % 5 T MacBook Pro JAO A Y & 7 U * 00 8 + - ( 9 S ) 0 1 0 PLet P0" (x) denote the quadratic polynomial that interpolates the data {(xo, yo), (x1, yı), (x2, y2)}; let P"."(x) denote the quadratic polynomial that interpolates the data {(x1, yı), (x2, y2), (x3, Y3)}. Finally, let P3(x) de- note the cubic polynomial interpolating the data{(xo, yo), (x1, yı), (x2, y2), (x3, y3)}. Show that 14. (a) (x3 – x)P0.2) (x) + (x – xo) P!.3) (x) P3 (x) = (1,3) X3 - Xo How might this be generalized to constructing P(x), interpolating {(xo, yo), (Xn, Yn)}, from interpolation polynomials of degreen – 1? (b) ....Let f(x) = x² + 2x – 1 and g(x)= x³ – 2x2 + 7. Compute a) (f) and (9). b) (G). (f ·g) and 2 (f o g).
- d Calculate r:(t) · r2(t)] and [ri(t) x r2(t)] first by differentiating dt the product directly and then by applying the formulas d [r1(t)· r2(t)] = r1(t) · dr2 , dri + dt r2(t) and dt dt dr2 dri ri(t) x r2(t)] = r¡(t) x dt x r2(t). dt dt r1(t) = 3ti + 3t²j + t°k, r2(t) = tʻk [r(t) r2(t)] d, [ri(t) × r2(t)] dtFind all non-negative integer pairs of (x, y) such that (xy − 7)² = x² + y²Find real numbers x, y and z such that (x + 4)i + (y – 5)j + (z – 1)k = 0