Determinants and Adjugate : Proof

Please see the attached file for the fully formatted problems. --- 1. Write a proof for the following statement: If A is any n x n non-singular matrix, then det(adj(A))=(det(A))^(n-1). Show work. Help: det: is the determinant adj: is the adjugate (or classical adjoint)

Vector spaces, linear transformation

1. Which of the following is not a linear transformation from R^3 to R^3?, explain why.

Vector Spaces: Subspace and Codomain

Write a proof for the following statement: The range of a linear transformation T:U->V is a subspace of the codomain V. Show work.

Vector Spaces : Rank

Please see the attached file for the full problem description. 1. Find the rank of A= [1 0 2 0] [ 4 0 3 0] [ 5 0 -1 0] [ 2 -3 1 1] . Show work. Help: [1 0 2 0] [ 4 0 3 0] [ 5 0 -1 0] : is a 4 x 4 matrices [ 2 -3 1 1]

Vector Geometry, Cross Products and Real Inner Products

Please see the attached file for full problem description. 1.Let u=(1,-1,3) and v=(2,-1,-1) be vectors in Euclidean 3-space R^3. Find a vector orthogonal to the plane of (subspace spanned by) the vectors u and v. Show work. Help: R^3: is Euclidean 3-space