Let u and v be nonzero orthogonal vectors in R3. Find bases for the range space and null space of the linear operator f:R3 into R3 defined by f(x) = uuTx + vvTx.

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

Let T : R3 → R3 be the orthogonal projection onto the line L in R3 containing all scalar multiples of the vector 1 (a) Express T in matrix form using a 3x3 matrix A. (b) Find the rank of the matrix A.

Let a be a fixed nonzero vector in R2. A mapping of the form L (x) = x + a is called a translation. Show that a translation is not a linear operator. Illustrate geometrically the effect of a translation.