Say you are given two vectors x and y represented using different basis sets {ê;} and {f¡}, of the same dimension N. That is, x = y = E yif;. AN. E x;e; and you are told that a given vector x has the representation T (1, 1)" in basis {ê;} and the representation x = {f}}. How might the two basis sets be related? Draw a figure to justify (-1,1)" in basis X = your answer.

12 If a. x and y are three non-zero and mutually non-parallel vectors, and the vectors b and e are given by b=x-(x-â)â and c=y-(y â)â-(y.b)b, where â and b are unit vectors along a and b respectively, show that a, b and c are mutually perpendicular.

1) Let a = 1 and b= 0 0 1 be two vectors. -1 What is |a|2|b|2+|ab|²?

-0---0 = 1. Which of these vectors is in the span of ₁, 2, 3? Justify your answers. For this entire problem set, consider the vectors #₁ = (a) -B (b) B -B (c) 2. (a) Describe the span of ₁, 2, 3 as a set of points in R³. (b) Is the span all of R³? 3. (a) Show that ws is in the span of ₁, ₂. and 3 = (b) Suppose 6 = 2w1 + ₂-33. Without calculating 6, show that b is in the span of w₁, 02. (c) Now calculate b using the given linear combination and using the linear combination you found. You should get the same vector.

§7. (6 p) Let each set S1, S2 be formed by only one vector: S₁ Is S₁ LI, LD or neither? Is S₂ LI, LD or neither? Is the set { , } LI, LD or neither? = { 7 }, S₂ = { 0