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: ...continues

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

Vector Geometry, Cross Products and Real Inner Products

Please see the attached file for the fully formatted problem. 1. Write the vector equation of the line in R^3 which passes through the two points P: (1,-1,3) and Q: (2,-1,-1). Show work. Help: R^3: is Euclidean 3-space

Vector Geometry, Cross Products and Real Inner Products: Moorean 3-Space

Please see the attached file for full problem description. 1. Demonstrate (check the properties) that the following function is an inner product in R^3. (Call R^3 with this inner product Moorean 3-space). Let u=(u_1,u_2,u_3) and v=(v_1,v_2,v_3). Then = uAv^T, where A=[ 2 0 0 ] ...continues

Vector geometry, cross products, and real inner products; Moorean 3-space

Please see the attached file for full problem description.

Vector geometry, cross products, and real inner products.

Please see the attached file for full problem description. 1. Let p_1=3-4x and p_2=3x+4x^2 be vectors (polynomials) in P_2. Use the integral inner product from 0 to 1 to find . Show work.

Vector geometry, cross products, and real inner products.

Please see the attached file for full problem description. Prove that if u and v are given non-zero vectors in the arbitrary inner-product space V, and are such that =0, then {u,v} is a linearly independent subset of V. Show work.