II Coordinates 5. Fix some a € R, and define polynomials Po(X), P1(X), P2(X) = P(R) by Po(X) = 1, p₁(X)=X+a, p₂(X) = (X + a)² (a) Prove that B = {po(X), P₁(X), p2(X)} is a basis of P₂ (R), the space of all polynomials of degree ≤ 2 with real coefficients. (b) For any f(X) = co + c₁X + c₂X² = P₂ (R), compute the coordinates of f with respect to the ordered basis B from part (a).

Q: Which of the following sets of polynomials form a basis for P₂? (i) P₁ = 1 + 4x + 6x², P2 3+x = (ii)…

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Q: Let P3 be the vector space of polynomial functions of degree less than or equal to 3 over the field…

A: Let P3 be the vector space of polynomial functions of degree less than or equal to 3 over the field…

Q: (2) Let T : R² → R´ and S : R´ → R³ be linear transformations defined by T x + y x2 2.x1 13 + 15 and…

A: The linear transformation is defined as T:ℝ2→ℝ5 and S:ℝ5→ℝ3, where Tx, yT=yxyx+yx-y and Sx1 x2 x3 x4…

Q: Let f(x) = −4, g(x) = -5x+7 and h(x) = - -2x29x+9. Consider the inner product (p,q) = p(-1)g(-1) +…

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Q: Consider the following ordered basis for P₂ (R): B {x²+2x+1, x + 1,1} [] 3 2 (a) What quadratic…

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Q: the basis of P2 cons tive to B.

A: Introduction: For some n, consider V to be a subspace of Rn. A basis for V is a collection of…

Q: Let P3 = {ao+a₁x + a₂x² + 3x³|ao, a₁, 02, 03 € R} be the vector space of polynomials of degrees at…

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Q: 2. In the vector space II3 [0, 1] of all polynomials on the unit interval [0, 1] of degree not…

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Q: Consider the basis B = {(2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for…

A: Given Data: Let us consider the given basis B =2,1,3,1 for V=R2 A=2311 is a matrix of basis is…

Q: a) Let q(t) = 4t2 + 2t + 9 be an element of P2. Write the coorfinate vector of q(t) relative to…

A: We have to find the followings ; (a) We have to find the coordinate vector of  q(t) = 4t2+2t+9 with…

Q: Define T:R-→R² by T(x) = Ax. Find a basis B for R² with the property that [T]B is diagonal. - 3 - 3…

A: Given:T:ℝ2→ℝ2 is defined by Tx=AxWhere A=-3-3-1-5To find basis B for ℝ2 such that TB is diagonal.…

Q: Computations: [7x -3y 9. Compute a basis of x, y E R x + y

A: We will find out the required basis.

Q: Consider the polynomials p₁ (t) = 2 + 5t, p₂ (t) = 2 − 5t, and p3 (t) = 4 (for all t). By…

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Q: 2.) Let Tv = Av represent the linear transformation T where 11 -1 2 1] 1 A Find a basis for (a) the…

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Q: (a) (b) (c) Let W be the subpace of M2x2 (R) b W = { [20 + $ 30+d]:akedER} a, b, c, Find a basis for…

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Q: Define the linear transformation T : P2 → P2 according to T (p(x)) = p(2x − 5). (a) Compute the…

A: Given T(p(x)) = p(2x − 5)

Q: 3) Let T d b }]= la + 1b + 1c + 5d + 2e +2f la + 2b + 0c + 5d + 3e + 1f 3a + 4b +3c + 17d+ 8e +6f…

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Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: asis: The set B = {2-3x², 8+x-12x², 33x² - (21+ 2x)} is a basis for P₂. Find the coordinates of p(x)…

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Q: The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z +…

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Q: `

A: Step 1: Step 2: Step 3: Step 4:

Q: Let f(x)=-4, g(x) = -5x+7 and h(x) = -2x²-9x + 9. Consider the inner product (p,q) = p(-1)q(−1) +…

A: Step 1:Step 2: Step 3: Step 4: