A quartic Bézier curve is determined by five control points, Po, P1, P2, P3, and p4: x(t) = (1 – t)*Po + 41(1 – t)°P¡ + 6t²(1 – t)²p2 + 41°(1 – t)P3 + t*P4 for 0 < t < 1 %3D Construct the quartic basis matrix MB for x(t).

1. (a) Express the polynomial q(t) = 3t² + 5t - 5 as a linear combination of the polynomials P₁ (t) = ² + 2t + 1, P2 (t) = 2t² + 5t + 4, P3(t) = 1² + 3t+ 6. (b) Consider the linear map T: R³ R³ such that T Y Z i. State the matrix of T in standard basis B of R³. ii. Determine the matrix of T in the basis C = = 2 {0)·()·()} - -1 y + z x + z y + x You may assume without proof that C forms a basis. 1 of R³.

Define T(f) = f′ from P4 to P4. (a) Prove T is linear. (b) Give the matrix for T. Is it invertible? Why or why not?

#10: Let W be the plane with the equation 3x-y + 62 = 0. Find the standard matrix P for the orthogonal projection onto W. Use the following formula P = A(ATA)-¹AT, where the matrix A is constructed using any basis for W as its column vectors. Enter the first row of the matrix P (in order) into the answer box below, separated with commas. i.e., enter the values P11, P12, P13.