"On R², define the operations of addition and scalar multiplication as follows: (x₁, x₂)+(V₁ Y2) = (x + y₂ +1, x₂ + y₂ + 1), c(x1, x₂) = (Cx: +c-1,c₂+c-1). Then R2 with these operations forms a vector space." Note that here the zero vector is (-1,-1) and the additive inverse of (x1, x2) is (-x₁-2, -x₂-2), not (-x₁-x₂) a) Show that (-1, -1) acts as the zero vector under these definitions.

Consider the following vectors: u = (1, −2, −1), v = (-2, 1, 8) Without using the cross product, find a vector n which is orthogonal to both u and U. You need to use a notion from chapter 5 to set up a system of linear equations, then use RREF to solve the system. Explain why this method works.

Let U be a subspace of R", and let S = { ū, v, w } be a set of three non-zero vectors in R”. Fill in the blanks so that the following statements are true. If you believe you don't have enough information to conclude anything for any of the cases, choose the answer "???". Note that there is no intended relationship between the parts. In each case, assume only what the statement says to assume. If SCU, then dim(U) > (Enter the largest integer that makes the statement necessarily true.) If S is linearly independent and S C U, then dim(U) 3. If S' is a spanning set for U, then dim(U) 3. If span(S) = U then S If S is a basis for U, then S linearly independent. linearly independent.

Calculate the scalar triple product u - (v x w), where u = (7,7,0), v = {-3, –9,7), and w = (7, –9, 1). (Use symbolic notation and fractions where needed.) u . (v x w) =

One of the following vectors lies in span{ 2+ x + x², 2x2} (A) 2+ 2x + 7x? (B) 2 + 2x + 5x? (C) 2 + 2x + 6x? (D) 2 + 2.x + 8x? (E) 2 + 2x + 9x2