A permutation operator, P, in a 3 dimensional linear vector space with basis vectors ê, ê, ê, has the following operation on the basis vectors: Pê₁ = ê3 Pê₂ = ê¹ Pê3 = ê₂. a) Derive P, the matrix form of the operator P, with the same basis used for the input and output spaces. Note: If you cannot solve this part then use the following matrix for the next two parts: Го о 17 100 10 b) Show that P3 = I, where I is the identity matrix. c) Hence show that for positive integers n and m: I for n P"=P P² for n = 3m = 3m +1 for n = 3m +2

Find a and that solve the vector equation: Suppose the above equation is expressed as Ax = b with x = coeffieints of the equations in one matrix form as (Ab)= Rows: 2 Columns: 3 b) c) d) 21 a (²) + ^ ( ;) = (²1) B α= B = Complete the missing column of the following matrix derived by doing row operation of the augmented matrix in a): 1 -2 (²). 0 1 Express the

8) Let A be an nxn ma triX. vector b the equation Ax = b has more than one solution. Explain why A is not an invertible Matrix.

If you keep the same basis vectors but put them in a different order, the change of basis matrix B is a __ matrix. If you keep the basis vectors in order but change their lengths, B is a __ matrix.

3= b, ,b2} and C= (c,.c2} be bases for a vector space V, and suppose b, = - 6c, + 5c2 and b2 = - 4c, +2c2. a. Find the change-of-coordinates matrix from B to C. b. Find (x]c for x = - 5b, + 3b2. Use part (a). a. P = C-B b. (xlc (Simplify your answers.)