Symmetric matrix

What is the symmetric matrix of the quadratic form 2x^2 - 8xy + 4y^2?

vectors

The set of vectors {[ 1 -1] , [ 1 -1] , [ 2 -1] } [ 2 0 ] [ -1 0] [ -1 0] from M_2(R) is: A. linearly dependent B. linearly independent C. orthogonal D. a spanning set for M_2(R) E. a basis for M_2(R)

Quadratic Equations

The graph of the quadratic equation x^T Ax = [0,0,1]x, where A = [ 1/(alpha^2) 0 0 ] [ 0 -1/(beta^2) 0 ] [ 0 0 0 ] is a(n): A. ellipse B. hyperbola C. elliptic paraboloid D. parabolic cone E. hyperbolic paraboloid

Conic graph

Identify and sketch the graph of the conic described by the quadratic equation x^2 + 4xy + y^2 - 12 = 0. Do this by writing this equation in matrix form; then change the equation to a sum of squares of the form x'^T Dx' where D is a diagonal matrix.

Matrix Representation of a Linear Transformation

For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Write the standard matrix representation , A = [T]_E, of T.

Eigenvalues

For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Find the eigenvalues of T.

Eigenvectors

For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Find the eigenvectors of T.

Change of Basis : Eigenvectors

For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Write the change of basis matrix K from the basis F of R^3 which consists of the eigenvectors of T to the standard basis E for R^3.

Diagonal matrix

For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). The matrix A = [T]_E is similar to a diagonal matrix D = [T]_F. Write the diagonal matrix D, and demonstrate that it is indeed similar to A by producing the appropriate non-singular matrix and its i ...continues

Kernel basis

For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). What is a basis for the kernel of T?