5. Find the function ax6 + bx5 + c which is closest to x on the interval [−1, 1]. Find an orthogonal basis of the space of polynomials {ax6+bx+c a, b, c = R}

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.

Determine if the vector product <A,B> = a0b0 defines a valid inner product on P2(R) for any A=a0+a1x+a2x^2 and B=b0+b1X+b2x2. Explain your answer clearly.

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

Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.

Assume that T is a linear transformation. T: Projects vectors in its domain onto the line y = 2/3 x. Find the standard matrix of T. Enter your matrix as 4-tuple (a11, 12, 921, 922) Use only fractions in simplest form and integers. NO DECIMALS "a11 11 "),a12= ,a21= ,a22=

Use matrix multiplication to find the orthogonal projection of (-8, 5, 7) on the xy -plane. Let T denote the linear operator mapping each vector into its orthogonal projection on the xy -plane. Then T(-8,5,7) = ( Ma )

Assume that T is a linear transformation. T: Projects vectors in its domain onto the line y = 4x. Find the standard matrix of T. Enter your matrix as 4-tuple (a11, 12, 021, 922) Use only fractions in simplest form and integers. NO DECIMALS "a11 11 "),a12= ,a21= ,a22=

Suppose T: R¹→P3 is a linear transformation whose action on a basis for R4 is as follows: -3 a C = II 9x²-12x+6 T 0 0 Describe the action of T on a general vector, using x as the variable for the polynomial and a, b, c, and d as constants. Use the '^' character to indicate an exponent, e.g. ax^2-bx+c. -4x²+4x-4 T −4x³−8x²+14x−14 T 2x³-6x²+5x-2

For those that you did not select as an element of P_2(R). Briefly explain why it was not an element of P_2(R).