Find the matrix A' = [LA]. Also find an invertible matrix Q such that A' = Q-¹AQ. (a) Suppose W₁ and W₂ are subspaces in a vector space V and that V = W₁ + W₂. Prove that V = W₁ W₂ if and only if any vector v € V can be represented uniquely as v = v₁ + v2 where V₁ € W₁, v₂ € W₂. (b) Suppose W₁ and W₂ are subspaces in a vector space V, and such that V =W₁ W₂. If B₁ is a basis of W₁ and 32 is a basis of W2 prove that 31 U 32 is a basis of V.

Find an orthogonal diagonalization for A = [[1, 1,0], [1, 1, 1], [0, 1, 1]], i. e. find an orthogonal matrix U and a diagonal matrix D such that U^ (TT)AU = D. Any empty entries are assumed to be 0.