Linear Algebra Answers

let V be a vector space of 3x3 symmetric matric.show that if A,B in V, then <A,B>=tr(AB) in inner product space

T(x₁,x₂,x₃) = (x₁₊x_2, x₂₊x_3, x_1-x_3, 2x₁+x₂-x₃) find he dimension of the kernel

prove that the eigen values of unitary matrix are unit modulus

Complete { (2, 0, 3)} to form an orthogonal 

basis of R³

Let : 

V = R3, 

W = {(X1, X2, X3) ! x1 - x2 = .X3}. 

Show that W is a subspace of V. Further, 

find a basis for W, and hence, find the 

dimension of W.

Let V be the vector space of polynomials of degree less than of equal to n.Show that the derivative operator on V is nilpotent of index n+1.

Let q(x₁, x₂)= 3x₁² - 6x₁x₂ +11x₂². Then find an orthonormal change of coordinate that diagonalizes the above quadratic form q

Find the adjoint of F:C³ tends to C³ defined by F(z_1, z_2, z₃)=(2z₁+(1-i)z₂ , (3+2i)z₁ -4iz₃ , 2iz₁+(4-3i)z₂ - 3z₃ )

Is the quadratic form x₁²+x₂²+2x₁x₃+4x₁x₃+3x₃² quadratic? Justify your answer

Find the symmetric matrix that corresponds to the quadratic equation 3x₁²+x₁x₃ -2x₂x₃