Show that the vectors u₁ = (3, 1,-4), = (3,1, –4), u₂ = (2, 5, 6), and u} = = (1, 4, 8) form a basis in R³. Find the coordinates of (1, 1, 1) in this basis.

Find the coordinate vector (v)s relative to the basis S= {v ,,v,,v;} 21 for v = (2, 1, -2), if V, = (2, 2, -1) V2 = (-1, -2, 1) V3 = (1, -1, 1)

The coordinate vector of the vector (-4,3,1) in the basis B = {u= (1,1,1); v = (1,0,1); w= (0,0,1) } is: O A. (1,-4,3) OB. (-4,3,1) OC. (3,- 7,5) O D. (2,-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.

Let the basis B = {b1, b2} and basis C = {c1, c2} where vectors b1 = (7 -2). b2 = 2 -1 , c1 = 4 1  c2= 5 2 Find the change of coordinates matrix from B to C. (b) Find the change of coordinates matrix from C to B.

Let a = (5, 6, −4) and b = (-3, 4, −7) be vectors. (A) Find the scalar projection of b onto a. Scalar Projection: (B) Decompose the vector b into a component parallel to a and a component orthogonal to a. Parallel component: ( Orthogonal Component: ( " " "