Change of Basis Set U = round(20 ∗ rand(4)) − 10, V = round(10 ∗ rand(4))...

Change of Basis Set

U = round(20 ∗ rand(4)) − 10,

V = round(10 ∗ rand(4))

and set b = ones(4, 1).

(a) We can use the MATLAB function rank to determine whether the column vectors of a matrix are linearly independent. What should the rank be if the column vectors of U are linearly independent? Compute the rank of U, and verify that its column vectors are linearly independent and hence form a basis for R⁴. Compute the rank of V, and verify that its column vectors also form a basis for R⁴.

(b) Use MATLAB to compute the transition matrix from the standard basis for R⁴ to the ordered basis E = {u₁, u₂, u₃, u₄}. [Note that in MATLAB the notation for the jth column vector uj is U(: , j ).] Use this transition matrix to compute the coordinate vector c of b with respect to E. Verify that

b = c₁u₁ + c₂u₂ + c₃u₃ + c₄u₄ = Uc

(c) Use MATLAB to compute the transition matrix from the standard basis to the basis F = {v₁, v₂, v₃, v₄}, and use this transition matrix to find the coordinate vector d of b with respect to F. Verify that

b = d₁v₁ + d₂v₂ + d₃v₃ + d₄v₄ = Vd

(d) Use MATLAB to compute the transition matrix S from E to F and the transition matrix T from F to E. How are S and T related? Verify that Sc = d and T d = c.