Let f(x) = −4, g(x) = -5x+7 and h(x) = - -2x29x+9. Consider the inner product (p,q) = p(-1)g(-1) + p(0)q(0) + p(1)q(1) in the vector space P2 of polynomials of degree at most 2. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of P2 spanned by the polynomials f(x), g(x) and h(x). {1 8 0 }.

Consider below presented polynomials v1, v2, v3, v4in P2. v1 = 2t2 + t + 2 v2 =t2 - 2t v3 =5t2 – 5t + 2 v4 =-t2 - 3t – 2 Determine whether the vector v = t2 + t + 2 belong to span {v1, v2, v3, v4}or not.

Use the inner product (f, 9) = | f(2)g(x) da in the vector space P(R) of polynomials to find (f, g), || f|| ||g||, and the angle afg between f(æ) and g(x) for 2 f(x) = 10x – 3 and g(x) = -9x + 10. (f, 9) = || F|| = l|g|| = afg

Let u1 = (1, 3, 2, 1), u2= (2, −2, −5, 4), u3 = (2, −1, 3, 6). If v = (2, 5, −4, 0), write v as a linear combination of u1, u2, u3. If it is not possible say so. Let V be the set of all fifth-degree polynomials with standard operations. Is it a vector space? Justify your answer.  Let V = {(x, y) : x ≥ 0, y ≥ 0} with standared operations. Is it a vector space. Justify your answer.  Let S = {(6, 2, 1),(−1, 3, 2)}. Determine, if S is linearly independent or dependent? Let S= {(1, 0, 0),(0, 4, 0),(0, 0, −6),(1, 5, −3)}. Determine, if S is linearly independent or dependent.

Let u1 = (1, 3, 2, 1), u2= (2, −2, −5, 4), u3 = (2, −1, 3, 6). If v = (2, 5, −4, 0), write v as a linear combination of u1, u2, u3. If it is not possible say so. Let V be the set of all fifth-degree polynomials with standard operations. Is it a vector space? Justify your answer.  Let V = {(x, y) : x ≥ 0, y ≥ 0} with standared operations. Is it a vector space. Justify your answer.

If A and B are orthogonal, is A+B also orthogonal?  Can you demonstrate Algebraically using A + B = (A + B)^T etc...? Thank you.

{ƒf(x) = P₂[x] |ƒ'(−8) =ƒ(1)} where P₂[x] is the vector space of polynomials in x with degree at most 2. You can enter polynomials using notation e.g., 5+3xx for 5 + 3x². p(x) = ,q(x) = =

T: P2 P2 defined by T(a + bx + cx²) = a + b(x + 8) + b(x + 8)² Find the following images of the function for vectors p₁(x) = a₁ + b₁x + c₁x² and p₂(x) = a₂ + b₂x + c₂x2 in P₂ and the scalar k. (Give all answers in terms of a₁, b₁, C₁, 22, b₂, C₂ and c.) T(P₁) + T(P₂): = T(P1 + P₂) = cT (P1) = T(CP1) = Determine whether T is a linear transformation. 4 Olinear transformation O not a linear transformation

Use the inner product (p, q) = agbo + a,b, + azb2 to find (p, 9), |P||, |9||, and d(p, q) for the polynomials in P2. p(x) = 1 - x + 2x2, g(x) = x - x2 (a) (p, a) (b) (c) |||| (d) d(p, q)