Let P be a fixed (but unknown) 3 x 2 matrix. Define a new subset Tp CM Tp = {A EM | AP = PA} i. Show that Tp is closed under addition ii. Show that Tp is closed under scalar multiplication iii. Determine if Tp is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace.

Let V₁ = (1,4,5,6,9), v₂ =(3, 2, 1,4, -1), V3 = (-1,0,-1, -2, -1), V4=(2,3,5,7,8), Find a basis for the orthogonal complement of the subspace of RS spanned by the vectors, V1, V2, V3, and V4. Choose the correct answer from the options below. a,((-1,-1,1,0,0), (-2,-1,0,1,0), (1, -2,0,0,1)) b,((-2,-1,1,0,0), (-2,-1,0, 1,0), (-1, -2,0,0,1)) c,(-1,-1,1,0,0), (-2,3,0,1,0), (-1, -2,0,0,1)) d,(-1,-1,1,0,0), (-2,-1,0,1,0), (-1,-1,0,0,1)).

Let W be the subspace of M₁0.10 (i.e., 10 × 10 matrices) consisting of all matrices whose diagonal entries are zero. Find the dimension of W.

if the set of vectors is a basis of R5. If not, determine the dimension of the subspace spanned by the vectors