1. Say whether the following vectors are linearly independent and whether the vectors form a basis of R³. Support your answers with reasons. (a) {(1,2,3), (-1,2,3)}. (b) (c) {(2,-2, 1), (0, 2, -2), (2, 0, -1)}. (d) {(1,2,3), (-1,2,3), (0, 0, 0)}. {(1, 1, 1), (1, −1, −1), (−1, 1, −1), (-1, -1, 1)}.

Either find the coordinates of the vector relative to basis B, or find the original vector from the coordinates relative to basis B. (a) v = (3,5,10); B = {(1,0,−1), (2, 1, −1), (-2, 1,4)} for R³. (b) v = (13,54); B = {(1,0), (0, 1)} for R². (c) [v]B = (10,-1, 4); B = {(1,0,−1), (2, 1, −1), (−2, 1,4)} for R³.

Find two vectors Vy and V2 whose sum is (-5, –3), where 7y is parallel to (2, 5) while Uz is perpendicular to (2, 5).

The following question is from linear algebra first year: Factors the vector (6, -5, -1)t into three components a,b,c that satisfy the following conditions: a depends on (2,0,1)t, b depends on (1,2, 0)t and c is orthogonal to a and b. Please show it step by step. Can we get integers as answers?

a. Write the vector (8, 5, 30) as a linear combination of a₁ = (4, -4,1), d₂ = (4, -5, 5) and d3 = (0, -4,-4). Express your answer in terms of the named vectors. Your answer should be in the form 4ả₁ +52 + 6ả3, which would be entered as 4a1 + 5a2 + 6a3. (8,5, 30) = b. Represent the vector (8,5,30) in terms of the ordered basis B = {(4,-4, 1), (4, −5, 5), (0, -4,-4)}. Your answer should be a vector of the general form . [(8, 5, 30)] B =

Let vi = (3,2,–-2) and v2 = (2,-1,-1). Which of the following vectors are in span{V1,V2}? (i) (1,3,0) (ii) (-1,4,0) (iii) (0,7,–1)