Linear Algebra Answers

Let W be a subspace of R^5, which is spanned by the vectors u₁ = (1, 2, 1, 0, 0) u₂=(0, 1, 3, 3, 1) u₃= (1, 4, 6, 4, 1)

Find a basis for W₀

Find the basis (a_1, a_2, a₃) that is dual to the following basis of R³

{u₁ = (1, -1, 3), u₂ = (0, 1, -1), u₃ = (0, 3, -2)}

Let B={u₁= (1, 1, -1) u₂= (1, 0, 1) u₃ = (3, 2, 0)} be a basis for R³. Find the dual basis of B

Find all possible rational canonical form for a 6×6 matrix over R with minimal polynomial m(t)= (t² -2t +3)(t+1)²

Given the vector u = (−1, 1, 2, 1) and the vectors v1 = (1, −1, 2, −1), v2 =

(−2, 2, 3, 2), v3 = (1, 2, 0, −1), v4 = (1, 0, 0, 1) which form an orthogonal ba-

sis for IR 4

, find the coordinate vector (u)S for the vector u. 

Diagonalise the matrix 𝐴 = [ 𝑎 𝑏 𝑐 𝑑 ], where 𝑎 + 𝑐 = 𝑏 + 𝑑, by finding a nonsingular matrix 𝑃 and a diagonal matrix 𝐷 such that 𝐴 = 𝑃𝐷𝑃 −1 . 

1 2 0 -1

2 6 -3 -3

3 10 -6 -5

Find the rank

(3 4 -1 1

1 _1 3 1

4 _3 11 2. Reduce to row reduced echelon form